Uploaded byshambhumath
180 views

PPT on Eigen values and Eigen vectors.pdf

The document focuses on eigenvalues and eigenvectors in linear algebra, detailing their geometrical interpretation, methods of determination, and the properties of related matrices such as symmetric and orthogonal matrices. It presents mathematical definitions, examples, and applications including how to calculate eigenvalues and eigenvectors for given matrices. Key concepts include diagonalization, algebraic and geometric multiplicity, and quadratic forms.

Embed presentation

Download to read offline
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Department of Mathematics College of Commerce, Arts & Science, Patna
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Outline Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Let A = [aij] be a given n × n matrix. Consider the vector equation Ax = λx. (1) Here x is an unknown vector and λ an unknown scalar. Our task is to determine x’s and λ’s that equation (1). Geometrically, solving eq. (1) in this way means that we are looking for vectors x for which the multiplication of x by the matrix A has the same effect as the multiplication of x by a scalar A, giving a vector Ax with components proportional to those of x, and A as the factor of proportionality.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Let A = [aij] be a given n × n matrix. Consider the vector equation Ax = λx. (1) Here x is an unknown vector and λ an unknown scalar. Our task is to determine x’s and λ’s that equation (1). Geometrically, solving eq. (1) in this way means that we are looking for vectors x for which the multiplication of x by the matrix A has the same effect as the multiplication of x by a scalar A, giving a vector Ax with components proportional to those of x, and A as the factor of proportionality. Clearly, the zero vector x = 0 is a solution of (1) for any value of λ, because A0 = 0 (this is of no interest).
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then Ax1 = 3 1 2 2 1 −2 = 1 −2 = x1.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then Ax1 = 3 1 2 2 1 −2 = 1 −2 = x1. Ax2 = 3 1 2 2 1 1 = 4 4 = 4x2. Thus x1 and x2 are the eigen vectors corresponding to eigen values λ1 = 1 and λ2 = 4.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A 1 −2 1 4 −1 0 = −1 −1 not an eigen vector of A 1 −2 1 4 −0.71 0.71 = 3 −0.71 0.71 is an eigen vector of A
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A 1 −2 1 4 −1 0 = −1 −1 not an eigen vector of A 1 −2 1 4 −0.71 0.71 = 3 −0.71 0.71 is an eigen vector of A 1 −2 1 4 −0.89 0.45 = 2 −0.89 0.45 is an eigen vector of A.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors The above observation leads us to the following definition. Definition Let A be any square matrix. A scalar λ is called an eignevalue of A if there exists a nonzero (column) vector x such that Ax = λx Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue λ .
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x. Remark: Any (nonzero) scalar multiple of an eigenvector is also a vector, i.e., A(kx) = k(Ax) = k(λx) = λ(kx)
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Q. How to find eigen values and eigen vectors?
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigenvalues and eigenvectors Q. Find the eigen values and eigen vectors of the matrix A = −5 2 2 −2 Solution. (a) Eigenvalues: Equation (1) is Ax = −5 2 2 −2 x1 x2 = λ x1 x2 ; in components, −5x1 + 2x2 = λx1 2x1 − 2x2 = λx2
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors I Transferring the terms on the right to the left. we get (−5 − λ)x1 + 2x2 = 0 (2) 2x1 + (−2 − λ)x2 = 0 (3) In matrix notation, (A − λI)x = 0 (4) Clearly, equation in (4) is a homogeneous linear system. If the homogeneous equation (4) has a non-trivial solution, then the coefficient matrix (A − λI) must be singular, i.e., determinant det(A − λI) of A is zero. det(A − λI) = −5 − λ 2 2 −2 − λ = λ2 + 7λ + 6 = 0. This quadratic equation in λ is called the characteristic equation of A. Its solution are the eigen values λ1 and λ2 of A. Here, λ1 = −1 and λ2 = −6.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors II (b1) Eigenvector of A corresponding to λ1: From equation (2), we get −4x1 + 2x2 = 0 2x1 − x2 = 0 A soution is x2 = 2x1. We choose x1 = 1, we get x2 = 2. Thus an eigen vector of A corresponding to λ1 = −1 is x1 = 1 2 . Check: Ax1 = −5 2 2 −2 1 2 = −1 −2 = (−1)x1 = λ1x1. (b2) Eigenvector of A corresponding to λ2: From equation (2), we get x1 + 2x2 = 0 2x1 + 4x2 = 0
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors III A soution is x2 = −x1/2. We choose x1 = 2, we get x2 = −1. Thus an eigen vector of A corresponding to λ2 = −6 is x2 = 2 −1 . Check: Ax2 = −5 2 2 −2 2 −1 = −12 6 = (−6)x2 = λ2x2. This example illustrates the general case (A is a square matrix of order n) as follows. Equation (1) written in components is a11x1 + a12x2 + ..... + a1nxn = λx1 a21x1 + a22x2 + ..... + a2nxn = λx2 ................................................ an1x1 + an2x2 + ..... + annxn = λxn
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors IV Transferring the tenns on the right side to the left side, we have (a11 − λ)x1 + a12x2 + ..... + a1nxn = 0 a21x1 + (a22 − λ)x2 + ..... + a2nxn = 0 ................................................ an1x1 + an2x2 + ..... + (ann − λ)xn = 0 In matrix notation, (A − λI)x = 0 (5) This homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero:
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors V det(A − λI) = a11 − λ a12 ... a1n a21 a22 − λ ... a2n .. .. .. .. an1 an2 ... ann − λ = 0 The above equation is called the characteristic equation of A and we obtain a polynomial of nth degree in λ, called the characteristic polynomial of A. Remark ▶ The eigenvalues of a square matrix A are the roots of the characteristic equation of A. ▶ An n × n matrix has at least one eigenvalue and at most n numerically different eigenvalues. Properties of eigenvalues and eigenvector
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors VI ▶ A and AT has the same eigenvalues. ▶ A−1 has the eigenvalue 1/λ and the corresponding eigenvector is x. ▶ For a real square matrix A, if α + iβ is an eigenvalue, then its conjugate α − iβ is also an eigenvalue. When A is complex then this property does not hold. For example: A = 0 1 −1 0
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples I We now give two example which illustrate that an n × n matrix may have n linearly independent eigenvectors, or it may have fewer than n. Q. Find the eigenvalues and eigenvectors of A =   4 1 −1 2 5 −2 1 1 2   Solution. The characteristic polynomial of A is λ3 − trace(A)λ2 + (A11 + A22 + A33)λ − det(A) = λ3 − 11λ2 + 39λ − 45 = (λ − 3)(λ2 − 8λ + 15) = (λ − 3)(λ − 5)(λ − 3) = (λ − 3)2(λ − 5). Thus λ = 3, 3, 5 are the eigen values of A. Eigenvector corresponding to λ = 3 are x1 =   1 −1 0  
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples II x2 =   1 0 1   Eigenvector corresponding to λ = 5 is x3 =   1 2 1   Thus X = {x1, x2, x3} = {[1 − 1 0]T, [1 0 1]T, [1 2 1]T} is a maximal set of linearly independent eigenvectors of A. Q. Find the eigen values and eigenvectors of A =   3 −1 1 7 −5 1 6 −6 2   Solution. The characteristic polynomial of A is λ3 − trace(A)λ2 + (A11 + A22 + A33)λ − det(A) = λ3 − 12λ − 16 = (λ − 2)2(λ + 4).
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples III Thus λ = 2, 2, −4 are the eigen values of A. Eigenvector corresponding to λ = 2 is x1 =   1 1 0   Eigenvector corresponding to λ = −4 is x3 =   0 1 1   Thus X = {x1, x2, x3} is a maximal set of linearly independent eigenvectors of A. Remark Eigenvectors of an n × n matrix A may (or may not) form a basis (basis of eigenvectors, called eigenbasis) for Rn. Definition
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples IV ▶ The order Mλof an eigenvalue λ as a root of the characteristic polynomial is called the algebraic multiplicity of λ. ▶ The number mλ of linearly independent eigenvectors corresponding to λ is called the geometric multiplicity of λ. ▶ The difference △λ = Mλ − mλ is called the defect of λ. Remark Thus mλ is the dimension of the eigenspace corresponding to this λ. Since the characteristic polynomial has degree n, the sum of all the algebraic multiplicities must equal n.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices I Definition A real square matrix A is called (i) symmetric if transposition leaves it unchanged, i.e., AT = A, (ii) skew-symmetric if transposition gives the negative of A, i.e., AT = −A, and (iii) orthogonal if transposition gives the inverse of A, i.e., AT = A−1. Example: The matrices A =   −3 1 5 1 0 −2 5 −2 4  , B =   0 9 −12 −9 0 20 12 −20 0  , C =   2/3 1/3 2/3 −2/3 2/3 1/3 1/3 2/3 −2/3  
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices II are symmetric, skew-symmetric, and orthogonal, respectively. Remark Every skew-symmetric matrix has all main diagonal entries zero. Any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S, where R = 1/2(A + AT) and s = 1/2(A − AT).
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices I Eigenvalues of Symmetric and Skew-Symmetric Matrices 1. The eigenvalues of a symmetric matrix are real. 2. The eigenvalues of a skew-symmetric matrix are pure imaginary or zero. Remark The determinant of an orthogonal matrix has the value +1 or −1. Eigenvalues of an Orthogonal Matrix The eigenvalues of an orthogonal matrix A are real or complex conjugates in pairs and have absolute value 1.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices I Definition: Basis of Eigenvectors If an n × n matrix A has n distinct eigenvalues, then A has a basis of eigenvectors x1, x2, ..., xn. Similar Matrices An n × n matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. In this case, the diagonal elements of D are the corresponding eigenvalues and D = P−1AP, where P is the matrix whose columns are the eigenvectors. Eigenvalues and Eigenvectors of Similar Matrices ▶ Similar matrices have same eigenvalues. ▶ If x is an eigenvector of A, then y = P−1x is an eigenvector of D corresponding to the same eigenvalue. For example: Let A = 6 −3 4 −1 P = 1 3 1 4
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices II Remark Also, Am = (PDP−1)m = PDmP−1 = P diag(λm 1 , λm 2 , ..., λm n )P−1, where λ1, λ2, ..., λn are eigen values of A. Further, if the diagonal entries of D are nonnegative, let B = P diag( √ λ1, √ λ2, ..., √ λn)P−1 Then B is a nonnegative square root of A; that is, B2 = A and the eigenvalues of B are nonnegative. Consider example 1: We observe that x1 and x2 are linearly independent and hence form a basis of R2. Accordingly, A is diagonalizable. Furthermore, let P be the matrix whose columns are the eigenvectors x1 and x2, i.e., P = 1 1 −2 1 and P−1 = 1/3 −1/3 2/3 1/3
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices III Then A is similar to the diagonal matrix D = P−1AP = 1/3 −1/3 2/3 1/3 3 1 2 2 1 1 −2 1 = 1 0 0 4 Thus matrix A can be written as A = PDP−1 = 1 1 −2 1 1 0 0 4 1/3 −1/3 2/3 1/3 Accordingly, we can find A4 = 1 1 −2 1 1 0 0 256 1/3 −1/3 2/3 1/3 = 171 85 170 86 Lastly, we obtain a positive square root of A. Specifically, using √ 1 = 1 and √ 4 = 2, we obtain the matrix B = 1 1 −2 1 1 0 0 2 1/3 −1/3 2/3 1/3 = 5/3 1/3 2/3 4/3 where B2 = A and B has positive eigenvalues 1 and 2.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices IV Now, consider the matrix A = 5 −1 1 3 We observe that λ = 4 is the only eigenvalue of A. The corresponding eigenvector is x = [1 1]T. Accordingly, A is not diagonalizable, since there does not exist a basis consisting of eigenvectors of A.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of real symmetric Matrices I Theorem Let A be a real symmetric matrix. Suppose x1 and x2 are eigenvectors of A belonging to distinct eigenvalues λ1 and λ2 Then x1 and x2 are orthogonal (vectors whose inner product is zero), that is, x1, x2 = 0. Remark Let A be a real symmetric matrix. Then there exists an orthogonal matrix P such that D = P−1AP is diagonal. The orthogonal matrix P is obtained by normalizing a basis of orthogonal eigenvectors of A. Consider the matrix A = 2 −2 −2 5
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes I A quadratic form Q in the components x1, x2, ..., xn of a vector x is a sum of n2 terms, i.e., Q = xTAX = Pn i=1 Pn j=1 aijxixj = a11x2 1 + a12x1x2 + ..... + a1nx1xn a21x2x1 + a22x2 2 + ..... + a2nx2xn ................................................ an1xnx1 + an2xnx2 + ..... + annx2 n A = [aij] is the coefficient matrix form. We may assume that A is symmetric, because we can take off-diagonal terms together in pairs and write the result as a sum of two equal terms. Remark
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes II As we know, the symmetric matrix has an orthonormal basis of eigenvectors. Hence if we take these as column vectors, we obtain a matrix X that is orthogonal, so that X−1 = XT. Thus we have, A = XDX−1 = XDXT. Then the quadratic form is Q = xT XDXT x (6) Let XTx = y. Then x = Xy, since XT = X−1. From equation (6), we have xTX = (XTX)T = yT and XTx = y, so that Q becomes Q = yT Dy = λ1y2 1 + λ2y2 2 + ... + λny2 n (7) The equation in (7) is known as principal axes form or canonical form, where λ1, λ2, ..., λn are the (not necessarily
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes III distinct) eigenvalues of the matrix A, and X is an orthogonal matrix with corresponding eigenvectors xx1, x2, ..., xn, respectively, as column vectors.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Application to quadratic form I Q. Find out what type of conic section the following quadratic form represents and transform it to principal axes: Q = 17x2 − 30xy + 17y2 = 128. Solution. We have Q = xTAx, where A = 17 −15 −15 17 x = x1 x2 The characteristic equation is given by (17 − λ)2 − 152 = 0. Thus λ1 = 2 and λ2 = 32. Q = 2y2 1 + 32y2 2 = 128 represents the ellipse, i.e., y2 1/82 + y2 2/22 = 1. If we want to know the direction1 of the principal axes in the xy-coordinates, we have to determine normalized eigenvectors from (A − λI)x = 0 with λ1 = 2 and λ2 = 32. Then
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Application to quadratic form II x1 = 1/ √ 2 1/ √ 2 x2 = −1/ √ 2 1/ √ 2 . Hence x = Xy = 1/ √ 2 −1/ √ 2 1/ √ 2 1/ √ 2 y1 y2 . x1 = y1/ √ 2 − y2/ √ 2 x2 = y1/ √ 2 + y2/ √ 2. This is a 45o rotation. 1 the directions of the position vector x of a point P(x, y) for which the direction of the position vector y of Q is the same or exactly opposite.
Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form VIT THANK YOU

More Related Content

PPTX
Diagonalization of matrix
byVANDANASAINI29
 
PPT
Eigen values and eigen vectors engineering
byshubham211
 
PPTX
Ravina773728o38363382836,230000603029.pptx
bysheetaljangra222
 
PPTX
matrices.pptx
byanonymus67
 
PPT
Lecture_06_Part_1.ppt
byYoganJayaKumar1
 
PPTX
Eigenvectors and Eigenvalues: A Comprehensive Essay Introduction Linear algeb...
byflashff3086
 
PPTX
eigenvalue
byAsyraf Ghani
 
PDF
151522-161226185046.pdf
bySatyabrataMandal7
 
Diagonalization of matrix
byVANDANASAINI29
 
Eigen values and eigen vectors engineering
byshubham211
 
Ravina773728o38363382836,230000603029.pptx
bysheetaljangra222
 
matrices.pptx
byanonymus67
 
Lecture_06_Part_1.ppt
byYoganJayaKumar1
 
Eigenvectors and Eigenvalues: A Comprehensive Essay Introduction Linear algeb...
byflashff3086
 
eigenvalue
byAsyraf Ghani
 
151522-161226185046.pdf
bySatyabrataMandal7
 

Similar to PPT on Eigen values and Eigen vectors.pdf

PPT
Eigenvalues and Eigenvectors
byVinod Srivastava
 
PPTX
Maths-->>Eigenvalues and eigenvectors
byJaydev Kishnani
 
PPTX
Eigen value and vector of linear transformation.pptx
byAtulTiwari892261
 
PPTX
Eigenvalue problems-numerical methods.pptx
byAnithaRaghavan3
 
PPTX
eigen_values.pptx
byPlanningHCEGC
 
PPT
LinearAlgebraReview.ppt
byShobhitTyagi46
 
PPTX
Linear algebra
bynagathangaraj
 
PPTX
Linear algebra
byanithaselvakumar271
 
PPT
Eigen values and eigen vectors
byRiddhi Patel
 
PPTX
Eigenvalue problems .ppt
bySelf-employed
 
PDF
Understanding Eigenvalues and Eigenvectors PPT presentation
byedusayandas
 
PPT
Maths
byVignesh Ravichandran
 
PDF
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...
byCeni Babaoglu, PhD
 
PPTX
Eigen values and eigen vectors
bytirath prajapati
 
PDF
Eigenvalues, Eigenvectors and Quadratic Forms.pdf
byAugustoMiguel Ramos
 
PDF
Eigenvalue eigenvector slides
byAmanSaeed11
 
PDF
Matlab eig
byAhmed Chegrani
 
PDF
Eigen value and vectors
byPraveen Prashant
 
PDF
Eigen-Decomposition: Eigenvalues and Eigenvectors.pdf
byNehaVerma933923
 
PPTX
151522-161226185046.pptx
bySatyabrataMandal7
 
Eigenvalues and Eigenvectors
byVinod Srivastava
 
Maths-->>Eigenvalues and eigenvectors
byJaydev Kishnani
 
Eigen value and vector of linear transformation.pptx
byAtulTiwari892261
 
Eigenvalue problems-numerical methods.pptx
byAnithaRaghavan3
 
eigen_values.pptx
byPlanningHCEGC
 
LinearAlgebraReview.ppt
byShobhitTyagi46
 
Linear algebra
bynagathangaraj
 
Linear algebra
byanithaselvakumar271
 
Eigen values and eigen vectors
byRiddhi Patel
 
Eigenvalue problems .ppt
bySelf-employed
 
Understanding Eigenvalues and Eigenvectors PPT presentation
byedusayandas
 
Maths
byVignesh Ravichandran
 
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...
byCeni Babaoglu, PhD
 
Eigen values and eigen vectors
bytirath prajapati
 
Eigenvalues, Eigenvectors and Quadratic Forms.pdf
byAugustoMiguel Ramos
 
Eigenvalue eigenvector slides
byAmanSaeed11
 
Matlab eig
byAhmed Chegrani
 
Eigen value and vectors
byPraveen Prashant
 
Eigen-Decomposition: Eigenvalues and Eigenvectors.pdf
byNehaVerma933923
 
151522-161226185046.pptx
bySatyabrataMandal7
 

Recently uploaded

PDF
Caribbean Examinations Council Literacy and Numeracy Standards
byCaribbean Examinations Council
 
PPTX
Statistical Data Analysis using R Programming.pptx
byVETRISELVIP1
 
PPTX
bundle of care.pptx Priti Bala @ BRD med
bypritibala13
 
PDF
GIÁO ÁN KẾ HOẠCH BÀI DẠY NĂNG LỰC SỐ MÔN TIẾNG ANH LỚP 12 CẢ NĂM - GLOBAL SUC...
byNguyen Thanh Tu Collection
 
PDF
Agents in Artificial Intelligence: Types, Architecture and Real-World Examples
byARTHIDEVARANIP
 
PDF
The Unique Wildlife of Ethiopia: From the Simien to the Bale Mountains
byfirtunagebremicheal
 
PDF
Problem Solving Agents in Artificial Intelligence with Examples and Water Jug...
byARTHIDEVARANIP
 
PDF
Types of Artificial Intelligence: Capabilities and Functionality Explained
byARTHIDEVARANIP
 
PPTX
Pneumonia, Social and Preventive Pharmacy .pptx
byVirat Ojha
 
PPTX
STERILITY INDICATOR Pharmaceutical microbiology
byChetana Wagh
 
PDF
McDowell Technical Community College Early Childhood Program Equitable Workfo...
byEducationNC
 
PPTX
ANTISEPTICS AND DISINFECTANTS CHAPTER NO.05.pptx
byGanu Gavade
 
PPTX
How to Manage Product Types in Odoo 18 Sales
byCeline George
 
PPTX
Exploring car engine oil and lubricants purpose and functions
bymjb77ny
 
PDF
NCA New Family Orientation 2026 Dosemagen.pdf
bytimdosemagen1
 
PPTX
Activity on Job position in Odoo 19 Recruitment
byCeline George
 
PPTX
How to create Article in Odoo 18 Knowledge App
byCeline George
 
PPTX
How to Manage Empty Location in Odoo 18 Inventory
byCeline George
 
PDF
RPT FORM 1 English (2026 academic session) SMKTS.pdf
byWANAHMADAZIMIBINWANA
 
PPTX
Nanomaterials and its types - Dr.M.Jothimuniyandi
byJothimuniyandi
 
Caribbean Examinations Council Literacy and Numeracy Standards
byCaribbean Examinations Council
 
Statistical Data Analysis using R Programming.pptx
byVETRISELVIP1
 
bundle of care.pptx Priti Bala @ BRD med
bypritibala13
 
GIÁO ÁN KẾ HOẠCH BÀI DẠY NĂNG LỰC SỐ MÔN TIẾNG ANH LỚP 12 CẢ NĂM - GLOBAL SUC...
byNguyen Thanh Tu Collection
 
Agents in Artificial Intelligence: Types, Architecture and Real-World Examples
byARTHIDEVARANIP
 
The Unique Wildlife of Ethiopia: From the Simien to the Bale Mountains
byfirtunagebremicheal
 
Problem Solving Agents in Artificial Intelligence with Examples and Water Jug...
byARTHIDEVARANIP
 
Types of Artificial Intelligence: Capabilities and Functionality Explained
byARTHIDEVARANIP
 
Pneumonia, Social and Preventive Pharmacy .pptx
byVirat Ojha
 
STERILITY INDICATOR Pharmaceutical microbiology
byChetana Wagh
 
McDowell Technical Community College Early Childhood Program Equitable Workfo...
byEducationNC
 
ANTISEPTICS AND DISINFECTANTS CHAPTER NO.05.pptx
byGanu Gavade
 
How to Manage Product Types in Odoo 18 Sales
byCeline George
 
Exploring car engine oil and lubricants purpose and functions
bymjb77ny
 
NCA New Family Orientation 2026 Dosemagen.pdf
bytimdosemagen1
 
Activity on Job position in Odoo 19 Recruitment
byCeline George
 
How to create Article in Odoo 18 Knowledge App
byCeline George
 
How to Manage Empty Location in Odoo 18 Inventory
byCeline George
 
RPT FORM 1 English (2026 academic session) SMKTS.pdf
byWANAHMADAZIMIBINWANA
 
Nanomaterials and its types - Dr.M.Jothimuniyandi
byJothimuniyandi
 

PPT on Eigen values and Eigen vectors.pdf

  • 1.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Linear Algebra Eigenvalues and Eigenvectors Shambhu Sharan Department of Mathematics College of Commerce, Arts & Science, Patna
  • 2.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Outline Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form
  • 3.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Let A = [aij] be a given n × n matrix. Consider the vector equation Ax = λx. (1) Here x is an unknown vector and λ an unknown scalar. Our task is to determine x’s and λ’s that equation (1). Geometrically, solving eq. (1) in this way means that we are looking for vectors x for which the multiplication of x by the matrix A has the same effect as the multiplication of x by a scalar A, giving a vector Ax with components proportional to those of x, and A as the factor of proportionality.
  • 4.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Let A = [aij] be a given n × n matrix. Consider the vector equation Ax = λx. (1) Here x is an unknown vector and λ an unknown scalar. Our task is to determine x’s and λ’s that equation (1). Geometrically, solving eq. (1) in this way means that we are looking for vectors x for which the multiplication of x by the matrix A has the same effect as the multiplication of x by a scalar A, giving a vector Ax with components proportional to those of x, and A as the factor of proportionality. Clearly, the zero vector x = 0 is a solution of (1) for any value of λ, because A0 = 0 (this is of no interest).
  • 5.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then
  • 6.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then Ax1 = 3 1 2 2 1 −2 = 1 −2 = x1.
  • 7.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors Example 1: Let A = 3 1 2 2 , x1 = 1 −2 , and x2 = 1 1 . Then Ax1 = 3 1 2 2 1 −2 = 1 −2 = x1. Ax2 = 3 1 2 2 1 1 = 4 4 = 4x2. Thus x1 and x2 are the eigen vectors corresponding to eigen values λ1 = 1 and λ2 = 4.
  • 8.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then
  • 9.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A
  • 10.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A
  • 11.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A 1 −2 1 4 −1 0 = −1 −1 not an eigen vector of A 1 −2 1 4 −0.71 0.71 = 3 −0.71 0.71 is an eigen vector of A
  • 12.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigen values and Eigen vectors Example 2: Consider a square matrix A = 1 −2 1 4 with some specific vectors. Then 1 −2 1 4 1 0 = 1 1 not an eigen vector of A 1 −2 1 4 0 1 = −2 4 not an eigen vector of A 1 −2 1 4 −1 0 = −1 −1 not an eigen vector of A 1 −2 1 4 −0.71 0.71 = 3 −0.71 0.71 is an eigen vector of A 1 −2 1 4 −0.89 0.45 = 2 −0.89 0.45 is an eigen vector of A.
  • 13.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Eigenvalues and Eigenvectors The above observation leads us to the following definition. Definition Let A be any square matrix. A scalar λ is called an eignevalue of A if there exists a nonzero (column) vector x such that Ax = λx Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue λ .
  • 14.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x.
  • 15.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x.
  • 16.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Geometrical interpretation of eigenvalues and eigenvectors x is an eigenvector of A if its image through A (i.e., Ax) is collinear with x. Remark: Any (nonzero) scalar multiple of an eigenvector is also a vector, i.e., A(kx) = k(Ax) = k(λx) = λ(kx)
  • 17.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Q. How to find eigen values and eigen vectors?
  • 18.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigenvalues and eigenvectors Q. Find the eigen values and eigen vectors of the matrix A = −5 2 2 −2 Solution. (a) Eigenvalues: Equation (1) is Ax = −5 2 2 −2 x1 x2 = λ x1 x2 ; in components, −5x1 + 2x2 = λx1 2x1 − 2x2 = λx2
  • 19.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors I Transferring the terms on the right to the left. we get (−5 − λ)x1 + 2x2 = 0 (2) 2x1 + (−2 − λ)x2 = 0 (3) In matrix notation, (A − λI)x = 0 (4) Clearly, equation in (4) is a homogeneous linear system. If the homogeneous equation (4) has a non-trivial solution, then the coefficient matrix (A − λI) must be singular, i.e., determinant det(A − λI) of A is zero. det(A − λI) = −5 − λ 2 2 −2 − λ = λ2 + 7λ + 6 = 0. This quadratic equation in λ is called the characteristic equation of A. Its solution are the eigen values λ1 and λ2 of A. Here, λ1 = −1 and λ2 = −6.
  • 20.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors II (b1) Eigenvector of A corresponding to λ1: From equation (2), we get −4x1 + 2x2 = 0 2x1 − x2 = 0 A soution is x2 = 2x1. We choose x1 = 1, we get x2 = 2. Thus an eigen vector of A corresponding to λ1 = −1 is x1 = 1 2 . Check: Ax1 = −5 2 2 −2 1 2 = −1 −2 = (−1)x1 = λ1x1. (b2) Eigenvector of A corresponding to λ2: From equation (2), we get x1 + 2x2 = 0 2x1 + 4x2 = 0
  • 21.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors III A soution is x2 = −x1/2. We choose x1 = 2, we get x2 = −1. Thus an eigen vector of A corresponding to λ2 = −6 is x2 = 2 −1 . Check: Ax2 = −5 2 2 −2 2 −1 = −12 6 = (−6)x2 = λ2x2. This example illustrates the general case (A is a square matrix of order n) as follows. Equation (1) written in components is a11x1 + a12x2 + ..... + a1nxn = λx1 a21x1 + a22x2 + ..... + a2nxn = λx2 ................................................ an1x1 + an2x2 + ..... + annxn = λxn
  • 22.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors IV Transferring the tenns on the right side to the left side, we have (a11 − λ)x1 + a12x2 + ..... + a1nxn = 0 a21x1 + (a22 − λ)x2 + ..... + a2nxn = 0 ................................................ an1x1 + an2x2 + ..... + (ann − λ)xn = 0 In matrix notation, (A − λI)x = 0 (5) This homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero:
  • 23.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors V det(A − λI) = a11 − λ a12 ... a1n a21 a22 − λ ... a2n .. .. .. .. an1 an2 ... ann − λ = 0 The above equation is called the characteristic equation of A and we obtain a polynomial of nth degree in λ, called the characteristic polynomial of A. Remark ▶ The eigenvalues of a square matrix A are the roots of the characteristic equation of A. ▶ An n × n matrix has at least one eigenvalue and at most n numerically different eigenvalues. Properties of eigenvalues and eigenvector
  • 24.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Determination of eigen values and eigen vectors VI ▶ A and AT has the same eigenvalues. ▶ A−1 has the eigenvalue 1/λ and the corresponding eigenvector is x. ▶ For a real square matrix A, if α + iβ is an eigenvalue, then its conjugate α − iβ is also an eigenvalue. When A is complex then this property does not hold. For example: A = 0 1 −1 0
  • 25.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples I We now give two example which illustrate that an n × n matrix may have n linearly independent eigenvectors, or it may have fewer than n. Q. Find the eigenvalues and eigenvectors of A =   4 1 −1 2 5 −2 1 1 2   Solution. The characteristic polynomial of A is λ3 − trace(A)λ2 + (A11 + A22 + A33)λ − det(A) = λ3 − 11λ2 + 39λ − 45 = (λ − 3)(λ2 − 8λ + 15) = (λ − 3)(λ − 5)(λ − 3) = (λ − 3)2(λ − 5). Thus λ = 3, 3, 5 are the eigen values of A. Eigenvector corresponding to λ = 3 are x1 =   1 −1 0  
  • 26.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples II x2 =   1 0 1   Eigenvector corresponding to λ = 5 is x3 =   1 2 1   Thus X = {x1, x2, x3} = {[1 − 1 0]T, [1 0 1]T, [1 2 1]T} is a maximal set of linearly independent eigenvectors of A. Q. Find the eigen values and eigenvectors of A =   3 −1 1 7 −5 1 6 −6 2   Solution. The characteristic polynomial of A is λ3 − trace(A)λ2 + (A11 + A22 + A33)λ − det(A) = λ3 − 12λ − 16 = (λ − 2)2(λ + 4).
  • 27.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples III Thus λ = 2, 2, −4 are the eigen values of A. Eigenvector corresponding to λ = 2 is x1 =   1 1 0   Eigenvector corresponding to λ = −4 is x3 =   0 1 1   Thus X = {x1, x2, x3} is a maximal set of linearly independent eigenvectors of A. Remark Eigenvectors of an n × n matrix A may (or may not) form a basis (basis of eigenvectors, called eigenbasis) for Rn. Definition
  • 28.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Examples IV ▶ The order Mλof an eigenvalue λ as a root of the characteristic polynomial is called the algebraic multiplicity of λ. ▶ The number mλ of linearly independent eigenvectors corresponding to λ is called the geometric multiplicity of λ. ▶ The difference △λ = Mλ − mλ is called the defect of λ. Remark Thus mλ is the dimension of the eigenspace corresponding to this λ. Since the characteristic polynomial has degree n, the sum of all the algebraic multiplicities must equal n.
  • 29.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices I Definition A real square matrix A is called (i) symmetric if transposition leaves it unchanged, i.e., AT = A, (ii) skew-symmetric if transposition gives the negative of A, i.e., AT = −A, and (iii) orthogonal if transposition gives the inverse of A, i.e., AT = A−1. Example: The matrices A =   −3 1 5 1 0 −2 5 −2 4  , B =   0 9 −12 −9 0 20 12 −20 0  , C =   2/3 1/3 2/3 −2/3 2/3 1/3 1/3 2/3 −2/3  
  • 30.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices II are symmetric, skew-symmetric, and orthogonal, respectively. Remark Every skew-symmetric matrix has all main diagonal entries zero. Any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S, where R = 1/2(A + AT) and s = 1/2(A − AT).
  • 31.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Symmetric, Skew-Symmetric, and Orthogonal Matrices I Eigenvalues of Symmetric and Skew-Symmetric Matrices 1. The eigenvalues of a symmetric matrix are real. 2. The eigenvalues of a skew-symmetric matrix are pure imaginary or zero. Remark The determinant of an orthogonal matrix has the value +1 or −1. Eigenvalues of an Orthogonal Matrix The eigenvalues of an orthogonal matrix A are real or complex conjugates in pairs and have absolute value 1.
  • 32.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices I Definition: Basis of Eigenvectors If an n × n matrix A has n distinct eigenvalues, then A has a basis of eigenvectors x1, x2, ..., xn. Similar Matrices An n × n matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. In this case, the diagonal elements of D are the corresponding eigenvalues and D = P−1AP, where P is the matrix whose columns are the eigenvectors. Eigenvalues and Eigenvectors of Similar Matrices ▶ Similar matrices have same eigenvalues. ▶ If x is an eigenvector of A, then y = P−1x is an eigenvector of D corresponding to the same eigenvalue. For example: Let A = 6 −3 4 −1 P = 1 3 1 4
  • 33.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices II Remark Also, Am = (PDP−1)m = PDmP−1 = P diag(λm 1 , λm 2 , ..., λm n )P−1, where λ1, λ2, ..., λn are eigen values of A. Further, if the diagonal entries of D are nonnegative, let B = P diag( √ λ1, √ λ2, ..., √ λn)P−1 Then B is a nonnegative square root of A; that is, B2 = A and the eigenvalues of B are nonnegative. Consider example 1: We observe that x1 and x2 are linearly independent and hence form a basis of R2. Accordingly, A is diagonalizable. Furthermore, let P be the matrix whose columns are the eigenvectors x1 and x2, i.e., P = 1 1 −2 1 and P−1 = 1/3 −1/3 2/3 1/3
  • 34.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices III Then A is similar to the diagonal matrix D = P−1AP = 1/3 −1/3 2/3 1/3 3 1 2 2 1 1 −2 1 = 1 0 0 4 Thus matrix A can be written as A = PDP−1 = 1 1 −2 1 1 0 0 4 1/3 −1/3 2/3 1/3 Accordingly, we can find A4 = 1 1 −2 1 1 0 0 256 1/3 −1/3 2/3 1/3 = 171 85 170 86 Lastly, we obtain a positive square root of A. Specifically, using √ 1 = 1 and √ 4 = 2, we obtain the matrix B = 1 1 −2 1 1 0 0 2 1/3 −1/3 2/3 1/3 = 5/3 1/3 2/3 4/3 where B2 = A and B has positive eigenvalues 1 and 2.
  • 35.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of Matrices IV Now, consider the matrix A = 5 −1 1 3 We observe that λ = 4 is the only eigenvalue of A. The corresponding eigenvector is x = [1 1]T. Accordingly, A is not diagonalizable, since there does not exist a basis consisting of eigenvectors of A.
  • 36.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Diagonalization of real symmetric Matrices I Theorem Let A be a real symmetric matrix. Suppose x1 and x2 are eigenvectors of A belonging to distinct eigenvalues λ1 and λ2 Then x1 and x2 are orthogonal (vectors whose inner product is zero), that is, x1, x2 = 0. Remark Let A be a real symmetric matrix. Then there exists an orthogonal matrix P such that D = P−1AP is diagonal. The orthogonal matrix P is obtained by normalizing a basis of orthogonal eigenvectors of A. Consider the matrix A = 2 −2 −2 5
  • 37.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes I A quadratic form Q in the components x1, x2, ..., xn of a vector x is a sum of n2 terms, i.e., Q = xTAX = Pn i=1 Pn j=1 aijxixj = a11x2 1 + a12x1x2 + ..... + a1nx1xn a21x2x1 + a22x2 2 + ..... + a2nx2xn ................................................ an1xnx1 + an2xnx2 + ..... + annx2 n A = [aij] is the coefficient matrix form. We may assume that A is symmetric, because we can take off-diagonal terms together in pairs and write the result as a sum of two equal terms. Remark
  • 38.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes II As we know, the symmetric matrix has an orthonormal basis of eigenvectors. Hence if we take these as column vectors, we obtain a matrix X that is orthogonal, so that X−1 = XT. Thus we have, A = XDX−1 = XDXT. Then the quadratic form is Q = xT XDXT x (6) Let XTx = y. Then x = Xy, since XT = X−1. From equation (6), we have xTX = (XTX)T = yT and XTx = y, so that Q becomes Q = yT Dy = λ1y2 1 + λ2y2 2 + ... + λny2 n (7) The equation in (7) is known as principal axes form or canonical form, where λ1, λ2, ..., λn are the (not necessarily
  • 39.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Quadratic form and Transformation to Principal Axes III distinct) eigenvalues of the matrix A, and X is an orthogonal matrix with corresponding eigenvectors xx1, x2, ..., xn, respectively, as column vectors.
  • 40.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Application to quadratic form I Q. Find out what type of conic section the following quadratic form represents and transform it to principal axes: Q = 17x2 − 30xy + 17y2 = 128. Solution. We have Q = xTAx, where A = 17 −15 −15 17 x = x1 x2 The characteristic equation is given by (17 − λ)2 − 152 = 0. Thus λ1 = 2 and λ2 = 32. Q = 2y2 1 + 32y2 2 = 128 represents the ellipse, i.e., y2 1/82 + y2 2/22 = 1. If we want to know the direction1 of the principal axes in the xy-coordinates, we have to determine normalized eigenvectors from (A − λI)x = 0 with λ1 = 2 and λ2 = 32. Then
  • 41.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form Application to quadratic form II x1 = 1/ √ 2 1/ √ 2 x2 = −1/ √ 2 1/ √ 2 . Hence x = Xy = 1/ √ 2 −1/ √ 2 1/ √ 2 1/ √ 2 y1 y2 . x1 = y1/ √ 2 − y2/ √ 2 x2 = y1/ √ 2 + y2/ √ 2. This is a 45o rotation. 1 the directions of the position vector x of a point P(x, y) for which the direction of the position vector y of Q is the same or exactly opposite.
  • 42.
    Linear Algebra Eigenvalues and Eigenvectors ShambhuSharan Eigenvalues and Eigenvectors Motivation Geometrical interpretation Determination of eigenvalues and eigenvectors Algebraic and geometric multiplicity, Defect Symmetric, Skew-Symmetric, and Orthogonal Matrices Diagonalization of Matrices Similar matrices Eigenvalues and eigenvectors of similar matrices Diagonalization of real symmetric Matrices Quadratic form Application to quadratic form principal axes form or canonical form VIT THANK YOU