1. The matrix is not invertible as it has repeated rows.
2. The eigenvalue is 0 since a matrix is not invertible if it has 0 as an eigenvalue.
3. The eigenvectors corresponding to 0 can be found by reducing the matrix A - 0I to row echelon form. This gives the equation x1 + x2 + x3 = 0 with x2 and x3 as free variables, so two linearly independent eigenvectors are (1, -1, 0) and (1, 0, -1).
1) An eigenvector of a square matrix A is a non-zero vector x that satisfies the equation Ax = λx, where λ is the corresponding eigenvalue.
2) The zero vector cannot be an eigenvector, but λ = 0 can be an eigenvalue.
3) For a matrix A, the eigenvectors and eigenvalues can be found by solving the system of equations (A - λI)x = 0, where λI is the identity matrix multiplied by the eigenvalue λ.
The document discusses eigenvalues, eigenvectors, and diagonalization of matrices. It begins by defining eigenvalues and eigenvectors and providing an example of finding them for a matrix. It then discusses computing eigenvalues and eigenvectors, including using the characteristic equation and polynomial. The document explains diagonalization of matrices, including when a matrix is diagonalizable. It provides examples of finding eigenvalues, eigenvectors, and diagonalizing symmetric matrices. It concludes by defining orthogonal matrices.
This document outlines topics related to matrices, including:
- Types of matrices such as real, square, row, column, null, sub, diagonal, scalar, unit, upper triangular, lower triangular, and singular matrices
- Characteristic equations, eigenvectors, and eigenvalues of matrices
- Properties of eigenvalues including that the sum of eigenvalues is the trace and the product is the determinant
- Examples of finding the sum and product of eigenvalues without directly calculating them
The document provides definitions and examples of key matrix concepts.
The document contains notes from a previous linear algebra class covering the following topics:
1. There will be a quiz tomorrow on sections 1.1-1.3 focusing on concepts rather than lengthy calculations.
2. Previous topics included systems of linear equations, row reduction, pivot positions, basic and free variables, and the span of vectors.
3. Determining if a vector is in the span of other vectors is equivalent to checking if the corresponding linear system is consistent.
4. Examples are provided of determining if homogeneous systems have non-trivial solutions based on the presence of free variables. The general solution of a homogeneous system is expressed in parametric vector form.
This document discusses several key linear algebra concepts:
1) A square matrix is diagonalizable if it can be transformed into a diagonal matrix through multiplication by an invertible matrix. Diagonalizable matrices can be easily raised to high powers.
2) Eigenvalues and eigenvectors are values and vectors that are unchanged by transformation by the matrix, up to a scaling factor for eigenvectors.
3) Orthogonal matrices preserve lengths and angles when multiplying vectors. The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation.
Eigenvalues and eigenfunctions are key concepts in linear algebra. An eigenfunction is a function that when operated on by a linear operator produces a constant multiplied version of itself. The constant is the corresponding eigenvalue. Eigenvalues are the solutions to the characteristic polynomial of the linear operator. Eigenfunctions are not unique as any constant multiple of an eigenfunction is also an eigenfunction with the same eigenvalue. The spectrum of an operator is the set of all its eigenvalues.
This document summarizes key concepts regarding eigenvalues and eigenvectors of matrices:
- Eigenvalues are scalars such that there exist non-zero eigenvectors satisfying Ax = λx.
- The characteristic equation states that λ is an eigenvalue if and only if it satisfies det(A - λI) = 0.
- A matrix is diagonalizable if it can be written as A = PDP-1, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors. Diagonalizable matrices can easily compute powers by raising the eigenvalues to powers.
1) An eigenvector of a square matrix A is a non-zero vector x that satisfies the equation Ax = λx, where λ is the corresponding eigenvalue.
2) The zero vector cannot be an eigenvector, but λ = 0 can be an eigenvalue.
3) For a matrix A, the eigenvectors and eigenvalues can be found by solving the system of equations (A - λI)x = 0, where λI is the identity matrix multiplied by the eigenvalue λ.
The document discusses eigenvalues, eigenvectors, and diagonalization of matrices. It begins by defining eigenvalues and eigenvectors and providing an example of finding them for a matrix. It then discusses computing eigenvalues and eigenvectors, including using the characteristic equation and polynomial. The document explains diagonalization of matrices, including when a matrix is diagonalizable. It provides examples of finding eigenvalues, eigenvectors, and diagonalizing symmetric matrices. It concludes by defining orthogonal matrices.
This document outlines topics related to matrices, including:
- Types of matrices such as real, square, row, column, null, sub, diagonal, scalar, unit, upper triangular, lower triangular, and singular matrices
- Characteristic equations, eigenvectors, and eigenvalues of matrices
- Properties of eigenvalues including that the sum of eigenvalues is the trace and the product is the determinant
- Examples of finding the sum and product of eigenvalues without directly calculating them
The document provides definitions and examples of key matrix concepts.
The document contains notes from a previous linear algebra class covering the following topics:
1. There will be a quiz tomorrow on sections 1.1-1.3 focusing on concepts rather than lengthy calculations.
2. Previous topics included systems of linear equations, row reduction, pivot positions, basic and free variables, and the span of vectors.
3. Determining if a vector is in the span of other vectors is equivalent to checking if the corresponding linear system is consistent.
4. Examples are provided of determining if homogeneous systems have non-trivial solutions based on the presence of free variables. The general solution of a homogeneous system is expressed in parametric vector form.
This document discusses several key linear algebra concepts:
1) A square matrix is diagonalizable if it can be transformed into a diagonal matrix through multiplication by an invertible matrix. Diagonalizable matrices can be easily raised to high powers.
2) Eigenvalues and eigenvectors are values and vectors that are unchanged by transformation by the matrix, up to a scaling factor for eigenvectors.
3) Orthogonal matrices preserve lengths and angles when multiplying vectors. The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation.
Eigenvalues and eigenfunctions are key concepts in linear algebra. An eigenfunction is a function that when operated on by a linear operator produces a constant multiplied version of itself. The constant is the corresponding eigenvalue. Eigenvalues are the solutions to the characteristic polynomial of the linear operator. Eigenfunctions are not unique as any constant multiple of an eigenfunction is also an eigenfunction with the same eigenvalue. The spectrum of an operator is the set of all its eigenvalues.
This document summarizes key concepts regarding eigenvalues and eigenvectors of matrices:
- Eigenvalues are scalars such that there exist non-zero eigenvectors satisfying Ax = λx.
- The characteristic equation states that λ is an eigenvalue if and only if it satisfies det(A - λI) = 0.
- A matrix is diagonalizable if it can be written as A = PDP-1, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors. Diagonalizable matrices can easily compute powers by raising the eigenvalues to powers.
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
The document discusses derogatory and non-derogatory matrices. A derogatory matrix is one where the degree of the minimal polynomial is less than the size of the matrix. A non-derogatory matrix has a minimal polynomial of the same degree as the size of the matrix. The document provides procedures to determine if a matrix is derogatory or non-derogatory by finding the characteristic equation, eigenvalues, and checking if the minimal polynomial annihilates the matrix. An example is provided to demonstrate determining a derogatory matrix and another for a non-derogatory matrix.
Partial midterm set7 soln linear algebrameezanchand
This document provides solutions to problems from Problem Set 7 in 18.06 Linear Algebra. It includes solutions to 6 problems involving eigenvalues and eigenvectors of matrices. Key details include:
- Finding eigenvalues and eigenvectors of specific matrices like A = [matrix]
- Showing that the characteristic polynomial of a matrix A equals 0 using its diagonalization
- Deriving that the inverse of an invertible matrix A can be written as a polynomial function of A
- Explaining that the eigenvalues of a matrix A are also the eigenvalues of its transpose AT, while the eigenvectors may differ.
The document discusses eigen-values and eigenvectors of matrices. It defines eigen-values as scalar values for which the equation Ax = λx has a non-trivial solution, and eigenvectors as the non-trivial solutions. It describes how to determine the eigen-values from the characteristic equation, and how to then determine the corresponding eigenvectors. It also discusses properties of eigen-values and provides an example calculation.
Eigenvalues and Eigenvectors (Tacoma Narrows Bridge video included)Prasanth George
- There is a quiz tomorrow on sections 3.1 and 3.2 of the course material. Calculators will not be allowed and determinants must be calculated using the methods learned.
- Eigenvalues and eigenvectors are related to the linear transformation of a matrix A acting on a vector x. They give a better understanding of the transformation.
- The 1940 collapse of the Tacoma Narrows Bridge is explained by oscillations caused by the wind frequency matching the bridge's natural frequency, which is the eigenvalue of smallest magnitude based on a mathematical model of the bridge. Eigenvalues are important for engineering structure design.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
The document discusses linear transformations and linear independence. It contains examples and explanations of:
1) How a matrix A can transform a vector x from R4 to a new vector b in R2, representing the linear transformation.
2) How finding vectors x such that Ax=b is equivalent to finding pre-images of b under the transformation A.
3) Key concepts related to linear transformations like domain and range.
These are the slides from the review session. THE FILE IS BIG AND MAY HAVE BEEN CORRUPTED. IF YOU CAN'T SEE IT THROUGH THE FLASH INTERFACE, JUST CLICK THE "DOWNLOAD" LINK and view it on your own computer.
The document defines linear independence and dependence of vectors and discusses some key properties:
- A set of vectors is linearly independent if the only solution to their linear combination equaling the zero vector is the trivial solution with all coefficients equal to 0.
- A set is linearly dependent if at least one vector can be written as a linear combination of the others.
- A set of one vector is independent if it is not the zero vector. A set of two vectors is dependent if one is a multiple of the other.
- If a set contains more vectors than the dimension of the vector space, the set must be dependent since there are more variables than equations.
Stability criterion of periodic oscillations in a (11)Alexander Decker
This document discusses ideals of the polynomial ring [ ]xFn
2
( )1mod −n
x and their applications to cyclic codes and error control in computer systems. It defines principal ideals and shows that every ideal of [ ]xFn
2
( )1mod −n
x is principal. It also proves theorems showing that the set of polynomials corresponding to a cyclic code C form an ideal, the generator polynomial g(x) of a cyclic code divides ∈−1n
x , and a polynomial is a codeword if and only if it is divisible by the generator polynomial g(x). The document concludes that principal ideals of cyclic codes can be used for optimal error detection, correction and
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
The document defines and explains key concepts related to functions including:
- Functions map elements from the domain to a range.
- The domain is the set of independent variables a function is defined for, which can be continuous or discrete.
- The range is the set of output values the function can take.
- Functions can have properties like being even, odd, continuous, increasing, decreasing, or periodic.
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
Numerical Methods - Power Method for Eigen valuesDr. Nirav Vyas
The document discusses the power method, an iterative method for estimating the largest or smallest eigenvalue and corresponding eigenvector of a matrix. It begins by introducing the power method and notes it is useful when a matrix's eigenvalues can be ordered by magnitude. It then provides the working rules for determining a matrix's largest eigenvalue using the power method, which involves iteratively computing the matrix-vector product and rescaling the vector. Finally, it includes an example applying the power method to estimate the largest eigenvalue and eigenvector of a 2x2 matrix.
This document discusses linear and quadratic functions. It begins by defining linear functions as having a constant slope and providing examples of linear relationships involving student grades and beer demand. It then discusses inverse linear functions and using linear functions to model tax rates. The document next discusses quadratic functions as having a u-shaped or hill-shaped graph depending on the coefficient of the x^2 term. It provides an example of solving a quadratic equation graphically and discusses how a quadratic equation can have 0, 1, or 2 solutions. The summary concludes by noting a special case where a quadratic function reduces to y=x^2.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
The document contains announcements for an upcoming exam:
1. Students should bring any grade related questions about quiz 2 without delay. Test 1 will be on February 1st covering sections 1.1-1.5, 1.7-1.8, 2.1-2.3 and 2.8-2.9.
2. A sample exam 1 will be posted by that evening. Students should review for the exam after the lecture.
3. The instructor will be available in their office all day the following day to answer any questions.
It also provides tips for preparing for the exam, including doing homework problems and sample exams within the time limit to practice time management.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
The document discusses derogatory and non-derogatory matrices. A derogatory matrix is one where the degree of the minimal polynomial is less than the size of the matrix. A non-derogatory matrix has a minimal polynomial of the same degree as the size of the matrix. The document provides procedures to determine if a matrix is derogatory or non-derogatory by finding the characteristic equation, eigenvalues, and checking if the minimal polynomial annihilates the matrix. An example is provided to demonstrate determining a derogatory matrix and another for a non-derogatory matrix.
Partial midterm set7 soln linear algebrameezanchand
This document provides solutions to problems from Problem Set 7 in 18.06 Linear Algebra. It includes solutions to 6 problems involving eigenvalues and eigenvectors of matrices. Key details include:
- Finding eigenvalues and eigenvectors of specific matrices like A = [matrix]
- Showing that the characteristic polynomial of a matrix A equals 0 using its diagonalization
- Deriving that the inverse of an invertible matrix A can be written as a polynomial function of A
- Explaining that the eigenvalues of a matrix A are also the eigenvalues of its transpose AT, while the eigenvectors may differ.
The document discusses eigen-values and eigenvectors of matrices. It defines eigen-values as scalar values for which the equation Ax = λx has a non-trivial solution, and eigenvectors as the non-trivial solutions. It describes how to determine the eigen-values from the characteristic equation, and how to then determine the corresponding eigenvectors. It also discusses properties of eigen-values and provides an example calculation.
Eigenvalues and Eigenvectors (Tacoma Narrows Bridge video included)Prasanth George
- There is a quiz tomorrow on sections 3.1 and 3.2 of the course material. Calculators will not be allowed and determinants must be calculated using the methods learned.
- Eigenvalues and eigenvectors are related to the linear transformation of a matrix A acting on a vector x. They give a better understanding of the transformation.
- The 1940 collapse of the Tacoma Narrows Bridge is explained by oscillations caused by the wind frequency matching the bridge's natural frequency, which is the eigenvalue of smallest magnitude based on a mathematical model of the bridge. Eigenvalues are important for engineering structure design.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
The document discusses linear transformations and linear independence. It contains examples and explanations of:
1) How a matrix A can transform a vector x from R4 to a new vector b in R2, representing the linear transformation.
2) How finding vectors x such that Ax=b is equivalent to finding pre-images of b under the transformation A.
3) Key concepts related to linear transformations like domain and range.
These are the slides from the review session. THE FILE IS BIG AND MAY HAVE BEEN CORRUPTED. IF YOU CAN'T SEE IT THROUGH THE FLASH INTERFACE, JUST CLICK THE "DOWNLOAD" LINK and view it on your own computer.
The document defines linear independence and dependence of vectors and discusses some key properties:
- A set of vectors is linearly independent if the only solution to their linear combination equaling the zero vector is the trivial solution with all coefficients equal to 0.
- A set is linearly dependent if at least one vector can be written as a linear combination of the others.
- A set of one vector is independent if it is not the zero vector. A set of two vectors is dependent if one is a multiple of the other.
- If a set contains more vectors than the dimension of the vector space, the set must be dependent since there are more variables than equations.
Stability criterion of periodic oscillations in a (11)Alexander Decker
This document discusses ideals of the polynomial ring [ ]xFn
2
( )1mod −n
x and their applications to cyclic codes and error control in computer systems. It defines principal ideals and shows that every ideal of [ ]xFn
2
( )1mod −n
x is principal. It also proves theorems showing that the set of polynomials corresponding to a cyclic code C form an ideal, the generator polynomial g(x) of a cyclic code divides ∈−1n
x , and a polynomial is a codeword if and only if it is divisible by the generator polynomial g(x). The document concludes that principal ideals of cyclic codes can be used for optimal error detection, correction and
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
The document defines and explains key concepts related to functions including:
- Functions map elements from the domain to a range.
- The domain is the set of independent variables a function is defined for, which can be continuous or discrete.
- The range is the set of output values the function can take.
- Functions can have properties like being even, odd, continuous, increasing, decreasing, or periodic.
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
Numerical Methods - Power Method for Eigen valuesDr. Nirav Vyas
The document discusses the power method, an iterative method for estimating the largest or smallest eigenvalue and corresponding eigenvector of a matrix. It begins by introducing the power method and notes it is useful when a matrix's eigenvalues can be ordered by magnitude. It then provides the working rules for determining a matrix's largest eigenvalue using the power method, which involves iteratively computing the matrix-vector product and rescaling the vector. Finally, it includes an example applying the power method to estimate the largest eigenvalue and eigenvector of a 2x2 matrix.
This document discusses linear and quadratic functions. It begins by defining linear functions as having a constant slope and providing examples of linear relationships involving student grades and beer demand. It then discusses inverse linear functions and using linear functions to model tax rates. The document next discusses quadratic functions as having a u-shaped or hill-shaped graph depending on the coefficient of the x^2 term. It provides an example of solving a quadratic equation graphically and discusses how a quadratic equation can have 0, 1, or 2 solutions. The summary concludes by noting a special case where a quadratic function reduces to y=x^2.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
The document contains announcements for an upcoming exam:
1. Students should bring any grade related questions about quiz 2 without delay. Test 1 will be on February 1st covering sections 1.1-1.5, 1.7-1.8, 2.1-2.3 and 2.8-2.9.
2. A sample exam 1 will be posted by that evening. Students should review for the exam after the lecture.
3. The instructor will be available in their office all day the following day to answer any questions.
It also provides tips for preparing for the exam, including doing homework problems and sample exams within the time limit to practice time management.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document defines eigenvalues and eigenvectors. An eigenvector is a non-zero vector whose direction does not change when a linear transformation is applied. The associated scalar multiplier is the eigenvalue. Eigenvalues are found by setting the determinant of A - λI equal to 0. This characteristic equation has roots that are the eigenvalues. Eigenvectors correspond to distinct eigenvalues and are nonzero solutions to (λI - A)x = 0. The document provides examples of finding eigenvalues and eigenvectors and lists several properties of eigenvalues and eigenvectors.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
The document summarizes key concepts related to systems of linear equations and linear algebra, including:
1) A system of n linear equations can be expressed in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector. If b = 0, the system is homogeneous, otherwise it is nonhomogeneous.
2) If the coefficient matrix A is nonsingular, the system Ax = b has a unique solution that can be found by computing x = A^-1b. If A is singular, the system may have no solution or infinitely many solutions.
3) A set of vectors is linearly dependent if there exist scalar multiples of
This document provides 18 problems related to matrices. The problems cover topics like finding eigen values and vectors, properties of eigen values under operations like inverse and powers, and applying Cayley-Hamilton theorem.
This document provides an introduction to eigen values and eigen vectors. It defines eigen values as scalar values for which the equation Ax = λx has a non-trivial solution, where A is a matrix and x is an eigenvector. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigen values. Methods for determining eigen values and eigen vectors are described, including using the characteristic polynomial and characteristic matrix. Properties of eigen values are outlined, and the Cayley-Hamilton theorem is explained, which states that any matrix satisfies its own characteristic equation. An example is provided to demonstrate calculating eigen values and vectors.
This document discusses eigen values, eigen vectors, and diagonalization of matrices. It defines eigen values as the roots of the characteristic equation of a matrix. Eigen vectors are non-zero vectors that satisfy AX=λX, where λ is the eigen value. Diagonalization is the process of transforming a matrix A into a diagonal matrix D using a similarity transformation with an invertible matrix P, such that D=P-1AP. The document provides examples to illustrate these concepts and lists various properties of eigen values and eigen vectors.
The document provides information about eigenvalues and eigenvectors. It begins by defining eigenvalues and eigenvectors, and how they relate to a matrix A. It then describes how to compute the eigenvalues and eigenvectors of a matrix by finding the characteristic polynomial and solving the characteristic equation. Two examples are provided to illustrate this process. The document also discusses eigenspaces and proves that the set of eigenvectors for a given eigenvalue forms a subspace. It introduces the concept of diagonalization of matrices using similarity transformations.
The document discusses the process for finding the eigenvalues of a square matrix. It begins by defining the characteristic equation as det(A - λI) = 0, where A is the matrix and λI subtracts λ from the diagonal. The characteristic polynomial is obtained by computing this determinant. For a 2x2 matrix, it is a quadratic equation that can be factored to find the two eigenvalues. Larger matrices may require numerical methods. The sum of eigenvalues equals the trace, and their product equals the determinant. A matrix will always have n eigenvalues for its size n. An example problem is presented to demonstrate the full process.
This document discusses eigen values and eigenvectors. It defines eigen values and eigenvectors as scalars (eigenvalues) and vectors (eigenvectors) that satisfy the equation Ax = λx, where A is a matrix and λ is an eigenvalue. It provides properties of eigenvalues, including that the sum of eigenvalues equals the trace of A. It also discusses algebraic and geometric multiplicity, the characteristic equation, Cayley-Hamilton theorem, and examples to illustrate these concepts.
Introduction to Artificial IntelligenceManoj Harsule
S: fuzzy relation defined on Y and Z.
To find the composite relation R o S on X and Z:
μR o S(x,z) = maxy [min(μR(x,y), μS(y,z))]
For each x and z, find the maximum membership grade obtained by considering all possible y values and taking the minimum of the membership grades of R and S.
This gives the generalized intersection-union definition of composition of fuzzy relations. It reduces to the usual composition rule when relations are crisp.
This document provides examples and explanations for solving various types of equations beyond linear and quadratic equations. These include polynomial equations, equations with fractional expressions, equations involving radicals, and equations of quadratic type. Step-by-step solutions are shown for sample equations of each type. Extraneous solutions are discussed. Applications involving dividing a lottery jackpot and calculating bird flight energy expenditure are presented.
1. The document provides notes from a linear algebra course, covering topics like matrix factorization, row reduction, column space, nullspace, and solving systems of equations.
2. Key concepts explained include LU, LDU, and row echelon factorizations of matrices. The column space and nullspace of a matrix are defined as important subspaces.
3. Solving systems of equations Ax=b is discussed, noting the solution set is the particular solution plus any vector in the nullspace. The system has a solution if and only if b is in the column space of A.
1. The document provides solutions to 15 problems involving matrix eigenvalues and characteristic equations. The problems cover finding the characteristic equation and eigenvalues of various matrices, properties of eigenvalues such as their sum and product, and relationships between a matrix and its characteristic equation.
2. Key ideas addressed include: the characteristic equation of an upper triangular matrix contains its diagonal elements as eigenvalues; the sum of eigenvalues equals the sum of diagonal elements; the product of eigenvalues can be used to find an unknown eigenvalue; and a matrix satisfies its own characteristic equation.
3. Methods demonstrated include finding the characteristic equation by calculating the determinant of A - λI, using properties of eigenvalues to solve for unknowns, and showing matrices meet their characteristic equations.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
This document discusses various matrix decomposition techniques including least squares, eigendecomposition, and singular value decomposition. It begins with an introduction to the importance of linear algebra and decompositions for applications. Then it provides examples of using least squares to fit curves to data and find regression lines. It defines eigenvalues and eigenvectors and provides examples of eigendecomposition. It also discusses diagonalization of matrices and using the eigendecomposition to raise matrices to powers. Finally, it discusses singular value decomposition and its applications.
This document provides an introduction to matrix algebra and random vectors. It defines key concepts such as vectors, matrices, matrix operations, and properties of positive definite matrices. Vectors are defined as arrays of real numbers that can be added or multiplied by scalars. Matrices are rectangular arrays of numbers that can be added or multiplied. Positive definite matrices are matrices where the quadratic form is always nonnegative. The eigenvalues and eigenvectors of a symmetric positive definite matrix allow geometric interpretation of distances defined by the matrix.
1) The document discusses algebraic expressions and operations involving terms, monomials, polynomials, binomials, trinomials, and rational expressions. It also covers evaluating expressions, adding, subtracting, multiplying and dividing algebraic expressions.
2) Procedures for solving equations, systems of equations, and inequalities are presented. This includes isolating variables, using substitution and elimination methods, solving quadratic and exponential equations, and determining the properties of roots.
3) Examples are provided to illustrate solving linear, quadratic and rational equations as well as solving and graphing inequalities.
The document contains announcements about an exam, practice exam, review sessions, and exam grading for a class. It states that Exam 2 will be on Thursday, February 25 in class. A practice exam will be uploaded by 2 pm that day. Optional review topics will be covered the next day but will not be on the exam. A review session will be held on Wednesday with office hours from 1-4 pm. It also reminds students that a different class starts on Monday and to collect graded exams on Friday between 7 am and 6 pm.
1. There will be a quiz on Quiz 4 after the next lecture. Exam 2 will be on Feb 25 and cover material from Exam 1 to what is covered on Feb 22.
2. A practice exam will be uploaded on Feb 22 after the remaining material is covered. Optional topics on Feb 23 will not be covered on the exam.
3. Review session on Feb 24 in class. Office hours on Feb 24 from 1-4pm.
- Quiz 4 will be tomorrow covering sections 3.3, 5.1, and 5.2 of the textbook. It will include 3 problems on Cramer's rule, finding eigenvectors given eigenvalues, and finding characteristic polynomials/eigenvalues of 2x2 and 3x3 matrices. Students must show all work.
- Chapter 6 objectives include extending geometric concepts like length, distance, and perpendicularity to Rn. These concepts are useful for least squares fitting of experimental data to a system of equations.
- The inner product of two vectors u and v in Rn is defined as their dot product, which is the sum of the component-wise products of corresponding elements in u and v.
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
1. The document announces that students should bring any exam 1 grade questions without delay, and that the homework for exam 2 has been uploaded and may be updated. It also notes that the last day to drop the class is February 4th and there is no class on that date.
2. The document covers topics from the last class including computing 3x3 determinants, determinants of triangular matrices, and techniques for larger matrices.
3. The document then provides examples of computing determinants and discusses important properties including that row operations do not change the determinant value while row interchanges flip the sign, and multiplying a row scales the determinant.
1. Quiz 3 will cover sections 3.1 and 3.2 on February 11th. No calculators will be allowed and determinants must be found using the methods taught.
2. The homework problems have been updated, so students should check for the latest list.
3. To find the inverse of a 3x3 matrix A, first find the adjugate of A (denoted adjA) by writing the cofactors with alternating signs, then divide adjA by the determinant of A.
The document contains announcements and information about an exam for a class. It includes the following key points:
- Students should bring any grade-related questions about Exam 1 without delay. The homework for Exam 2 has been uploaded.
- The professor is planning to cover chapters 3, 5, and 6 for Exam 2.
- The last day for students to drop the class with a grade of "W" is February 4th.
The document contains announcements and information about an upcoming exam:
- A quiz and test are scheduled. Sample exams and review sessions will be provided.
- Exam 1 will cover several sections of the textbook and the professor will be available for questions.
- Tips are provided for studying including doing homework, examples, and practicing sample exams.
- Sections about subspaces and column/null spaces of matrices are summarized, including properties and examples.
Quiz 2 will be held on January 27 covering sections 1.4, 1.5, 1.7, and 1.8. Test 1 is scheduled for February 1. The document then provides steps to find the inverse of a 2x2 matrix, discusses invertibility if the determinant is 0, and gives an example of finding the inverse of a 3x3 matrix using row reduction of the augmented matrix.
The document discusses the following:
1. There will be a quiz on Jan 27 covering sections 1.4, 1.5, 1.7, and 1.8 and any issues with quiz 1 should be discussed asap.
2. Test 1 will be on Feb 1 in class with more details to come.
3. Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second matrix.
Quiz 2 will cover sections 1.4, 1.5, 1.7, and 1.8 on Wednesday January 27. Students with issues on quiz 1 should discuss with the instructor as soon as possible. The solution to quiz 1 will be posted on the website by Monday.
The document discusses linear transformations and provides examples of applying linear transformations to vectors. It defines key concepts such as the domain, co-domain, and range of a transformation. Examples are provided of interesting linear transformations including rotation and reflection transformations. Solutions to examples involving finding the image of vectors under given linear transformations are shown.
- There will be no class on Monday for Martin Luther King Day.
- Quiz 1 will be held in class on Wednesday and will cover sections 1.1, 1.2, and 1.3.
- Students should know all definitions clearly for the quiz, which will focus on conceptual understanding rather than lengthy calculations.
The document contains announcements and information about a class. It announces corrections to lecture slides, the last day to drop the class with a refund, and provides definitions and examples related to echelon form, reduced row echelon form, pivot positions, and solving systems of linear equations.
The document contains announcements from a class instructor. It notifies students that if they have not been able to access the class website or did not receive an email, to contact the instructor. It also reminds students that homeworks are posted on the class website and to check for any updates.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Eigenvalues - Contd
1. Announcements
Quiz 3 after lecture.
Grades will be updated online (including quiz 3 grades) over
the weekend. Please let me know if you spot any mistakes.
Exam 2 will be on Feb 25 Thurs in class. Details later.
Make-up exams will be given only if there is an excused
absence from the Dean of Students or a Doctor's note about
sudden serious illness. No exceptions on this. Travel
plans/broken alarm clock are unacceptable excuses.
2. Last Class...
Denition
An eigenvector of an n × n matrix A is a NON-ZERO vector x
such that Ax = λx for some scalar λ.
A scalar λ is called an eigenvalue of A if there is a nontrivial (or
nonzero) solution x to Ax = λx; such an x is called an eigenvector
corresponding to λ.
3. Triangular Matrices
Theorem
The eigenvalues of a triangular matrix are the entries on its main
diagonal.
4. Triangular Matrices
Example
1. Let
5 1 9
A = 0 2 3
0 0 6
The eigenvalues of A are 5, 2 and 6.
5. Triangular Matrices
Example
1. Let
5 1 9
A = 0 2 3
0 0 6
The eigenvalues of A are 5, 2 and 6.
2. Let
4 1 9
A= 0 0 3
0 0 6
The eigenvalues of A are 4, 0 and 6.
6. Zero Eigenvalue??
Zero eigenvalue means
1. The equation Ax = 0x has a nontrivial or nonzero solution
7. Zero Eigenvalue??
Zero eigenvalue means
1. The equation Ax = 0x has a nontrivial or nonzero solution
2. This means Ax = 0 has a nontrivial solution.
8. Zero Eigenvalue??
Zero eigenvalue means
1. The equation Ax = 0x has a nontrivial or nonzero solution
2. This means Ax = 0 has a nontrivial solution.
3. This means A is not invertible (or det A = 0) (by invertible
matrix theorem)
9. Zero Eigenvalue??
Zero eigenvalue means
1. The equation Ax = 0x has a nontrivial or nonzero solution
2. This means Ax = 0 has a nontrivial solution.
3. This means A is not invertible (or det A = 0) (by invertible
matrix theorem)
Zero is an eigenvalue of A if and only if A is not invertible
10. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
11. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
12. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Multiply both sides by A. We get A2 x = A(λx).
13. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Multiply both sides by A. We get A2 x = A(λx).
This is same as writing A2 x = λ (Ax) since λ is a scalar.
14. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Multiply both sides by A. We get A2 x = A(λx).
This is same as writing A2 x = λ (Ax) since λ is a scalar.
Again Ax = λx. So, A2 x = λ (λx) = λ2 x.
15. Important
If λ is an eigenvalue of a square matrix A, prove that λ2 is an
eigenvalue of A2 .
For any problem of this type, start with the equation that denes
eigenvalue and take it from there.
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Multiply both sides by A. We get A2 x = A(λx).
This is same as writing A2 x = λ (Ax) since λ is a scalar.
Again Ax = λx. So, A2 x = λ (λx) = λ2 x. This equation means that
λ2 is an eigenvalue of A2 .
16. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
17. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
18. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Since A is invertible, multiply both sides by A−1 . We get
−1
A A x = A−1 (λx)
I
19. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Since A is invertible, multiply both sides by A−1 . We get
−1
A A x = A−1 (λx) =⇒ A−1 (λx) = x
I
20. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Since A is invertible, multiply both sides by A−1 . We get
−1
A A x = A−1 (λx) =⇒ A−1 (λx) = x
I
This is same as writing λ(A−1 x) = x since λ is a scalar.
21. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Since A is invertible, multiply both sides by A−1 . We get
−1
A A x = A−1 (λx) =⇒ A−1 (λx) = x
I
This is same as writing λ(A−1 x) = x since λ is a scalar.
Since λ = 0 (Why?) we can divide both sides by λ and we get
−1 1
A x = λ x.
22. Important, see prob 25 sec 5.1
If λ is an eigenvalue of an invertible matrix A, prove that λ−1 is an
eigenvalue of A−1 .
Solution: If λ is an eigenvalue of a square matrix A, we have
Ax = λx.
Since A is invertible, multiply both sides by A−1 . We get
−1
A A x = A−1 (λx) =⇒ A−1 (λx) = x
I
This is same as writing λ(A−1 x) = x since λ is a scalar.
Since λ = 0 (Why?) we can divide both sides by λ and we get
−1 1
A x = λ x.
1
Thus λ or λ−1 is an eigenvalue of A−1 .
23. Example 20 section 5.1
Without calculation, nd one eigenvalue and 2 linearly independent
5 5 5
eigenvectors of A = 5 5 5 . Justify your answer.
5 5 5
Solution: What is special about this matrix? Invertible/Not
Invertible?
24. Example 20 section 5.1
Without calculation, nd one eigenvalue and 2 linearly independent
5 5 5
eigenvectors of A = 5 5 5 . Justify your answer.
5 5 5
Solution: What is special about this matrix? Invertible/Not
Invertible?
Clearly not invertible (same rows, columns).
What is an eigenvalue of A?
25. Example 20 section 5.1
Without calculation, nd one eigenvalue and 2 linearly independent
5 5 5
eigenvectors of A = 5 5 5 . Justify your answer.
5 5 5
Solution: What is special about this matrix? Invertible/Not
Invertible?
Clearly not invertible (same rows, columns).
What is an eigenvalue of A?0!!
To nd eigenvectors for this eigenvalue, we look at A − 0I and row
reduce. Or row reduce A.
26. Example 20 section 5.1
We get (do the row reductions yourself)
1 1 1 0
0 0 0 0
0 0 0 0
.
27. Example 20 section 5.1
We get (do the row reductions yourself)
1 1 1 0
0 0 0 0
0 0 0 0
. Thus x1 + x2 + x3 = 0 where x2 and x3 are free. So x1 = −x2 − x3 .
28. Example 20 section 5.1
We get (do the row reductions yourself)
1 1 1 0
0 0 0 0
0 0 0 0
. Thus x1 + x2 + x3 = 0 where x2 and x3 are free. So x1 = −x2 − x3 .
We have
−1 −1
x1 −x2 − x3
x2 = x2 = x2 1 + x3 0
x3 x3 0 1
29. Example 20 section 5.1
We get (do the row reductions yourself)
1 1 1 0
0 0 0 0
0 0 0 0
. Thus x1 + x2 + x3 = 0 where x2 and x3 are free. So x1 = −x2 − x3 .
We have
−1 −1
x1 −x2 − x3
x2 = x2 = x2 1 + x3 0
x3 x3 0 1
Choose x2 = 1 and x3 = 1 (or anything nonzero) and two linearly
independent eigenvectors are
−1 −1
1 , 0
0 1
30. Observations
1. The eigenvalue λ = 0 has 2 linearly independent eigenvectors
31. Observations
1. The eigenvalue λ = 0 has 2 linearly independent eigenvectors
2. We say that this eigenspace is a two-dimensional subspace of
R3 .
32. Observations
1. The eigenvalue λ = 0 has 2 linearly independent eigenvectors
2. We say that this eigenspace is a two-dimensional subspace of
R3 .
3. Examples with 2 linearly independent eigenvectors are very
important for section 5.3 when we do diagonalization and in
dierential equations where eigenvalues repeat.
39. Theorem
Theorem
Eigenvectors corresponding to distinct eigenvalues of an n ×n
matrix A are linearly independent.
40. Next week...
1. How to nd eigenvalues of a 2 × 2 and 3 × 3 matrix?
2. The process of diagonalization (uses eigenvalues and
eigenvectors)
3. Finding complex eigenvalues
4. Quick look at eigenvalues and eigenvectors being used to learn
long-term behavior of a dynamical system.
5. Quiz 4 (last quiz) will be on thurs Feb 18 based on sections
3.3, 5.1 and 5.2.