CONTENTS• Matrices and Linear systems of equations• Eigen values and eigen vectors• Real and complex matrices and Quadratic forms• Algebraic equations transcendental equations and Interpolation• Curve Fitting, numerical differentiation & integration• Numerical differentiation of O.D.E• Fourier series and Fourier transforms• Partial differential equation and Z-transforms
TEXT BOOKS• 1.Mathematical Methods, T.K.V.Iyengar, B.Krishna Gandhi and others, S.Chand and company• Mathematical Methods, C.Sankaraiah, V.G.S.Book links.• A text book of Mathametical Methods, V.Ravindranath, A.Vijayalakshmi, Himalaya Publishers.• A text book of Mathametical Methods, Shahnaz Bathul, Right Publishers.
REFERENCES• 1. A text book of Engineering Mathematics, B.V.Ramana, Tata Mc Graw Hill.• 2.Advanced Engineering Mathematics, Irvin Kreyszig Wiley India Pvt Ltd.• 3. Numerical Methods for scientific and Engineering computation, M.K.Jain, S.R.K.Iyengar and R.K.Jain, New Age International Publishers• Elementary Numerical Analysis, Aitkison and Han, Wiley India, 3rd Edition, 2006.
UNIT HEADER Name of the Course:B.Tech Code No:07A1BS02Year/Branch:I Year CSE,IT,ECE,EEE,ME, Unit No: III No.of slides:27
UNIT INDEX UNIT-IIIS.No. Module Lecture PPT Slide No. No. 1 Quadratic Form, L1-5 8-22 Reduction to canonicl form. 2 Real Matrices L6-7 23-25 3 Complex Matices L8-10 25-27
LECTURE-1Quadratic form: A homogeneous polynomial ofdegree two in any no.of variables is known as “quadratic form” Ex: 1).2x2+4xy+3y2 is a quadratic form in two variables x and y 2).x2-4y2+2xy+6z2-4xz+6yz is a quadratic form in three variables x,y and zGeneral quadratic form: The general quadratic form in n nn variables x1,x2,x3…………xn is defined as ∑∑ a x x i =1 j =1 ij i jWhere aij ‘s are constants.If aij ‘s are real then quadratic form is known as real quadratic form
Matrix of a quadratic form: The general quadratic form n n ∑∑ a x x i =1 j =1 ij i j where aij=aji can always be writtenas XTAX where x 1 [x x . . . x ] x 2 X= . , XT= 1 2 n . x n a 11 a 12 . . . . a1n The symmetric matrix A= [aij] = a 21 a 22 . . . . a2n . . a n1 a n 2 . . . . a nn is called the matrix of the quadratic form X TAX
NOTE: 1.The rank r of the matrix A is called the rank of the quadratic form XTAX 2.If the rank of A is r < n ,no.of unknowns then the quadratic form is singular otherwise non-singular and A=AT3. Symmetric matrix ↔ quadratic form
LECTURE-2 Nature,Index,Rank and signature of the quadratic fun: Let XTAX be the given Q.F then it is said to be Positive definite if all the eigen values of A are +ve Positive semi definite if all the eigen values are +ve and at least one eigen value is zero Negative definite if all the eigen values of A are –ve Negative semi definite if all the eigen values of A are –ve and at least one eigen value is zero Indefinite if some eigen values are +ve and some eigen values are -ve
Rank of a Q.F: The no.of non-zero terms in the canonical form of a quadratic function is called the rank of the quadratic funcand it is denoted by rIndex of a Q.F: Index is the no.of terms in the canonical form.It is denoted by p.Signature of a Q.F: The difference between +ve and –veterms in the canonical form is called the signature of the Q.F.And it is denoted by s Therefore, s = p-(r-p) = 2p-r where p = index r = rank
LECTURE-3Method of reduction of Q.F to C.F: A given Q.F can be reduced to a canonical form(C.F) by using the following methods1.by Diagonalization2.by orthogonal transformation or Orthogonalization3.by Lagrange’s reduction
1. Given a Q.F. reduces to the matrix form2. Find the eigen values3. Write the spectral matrix D = λ 1 0 0 4. 0 λ 2 0 Canonical form is YTDY where Y= 0 0 λ 3 y 1 y 2 C.F = y3 λ1 0 0 y1 =[ y 1 y y] 2 3 0 λ 2 0 y 2 0 0 λ 3 y 3 y 1 [y λ 1 1 yλ2 2 ] y3 λ 3 y 2 y3 2 y1 λ +y λ +y λ 1 2 2 2 2 3 3
LECTURE-4Method 2: Orthogonal transformation Write the matrix A of the Q.F Find the eigen values λ1,λ2,λ3 and corresponding eigen vectors X1,X2,X3 in the normalized form i.e.,||X1||,||X[2||,||X3|| ] e e e 1 2 3 Write the model matrix B= formed by normalized vectors . Where ei=Xi/||Xi|| B being orthogonal matrix B-1=BT so that BTAB=D,where D is the diagonal matrix formed by eigen values. y12 λ 1 + y22 λ 2 + y32 λ 3 The canonical form YT(BTAB)Y = YTDY = The orthogonal transformation X=BY
LECTURE-5Method 3: Lagrange’s reduction Take the common terms from product terms of given Q.F Make perfect squares suitable by regrouping the terms The resulting relation gives the canonical form
LECTURE-7Complex matrices: If the elements of a matrix, then the matrix is called a complex matrix. 1 + i i − 2 − 2 + i is a complex matrix Conjugate matrix: If A=[aij]mxn is a complex matrix then conjugate of A is A=[aij]mxn 1 + i 2i 1 − i − 2i 0 3 + 6i 0 then A= 3 − 6i
Conjugate transpose: conjugate transpose of a matrix A A is ( )T2= iA3i 2i + ө 2 − i − 3i − 2i A i 6 − 2i 9 − i 6 + 2i 9 A= , = 2− i −i − 3i 6 − 2i − 2i 9 Then Aө = k Note: 1. (Aө)ө = A 2. (kA)ө = Aө , k is a complex number 3. (A+B)ө = Aө + Bө
aijHermitian matrix: A square matrix A=[aij] is said to be hermitian if aij =aji- for all i and j. The diagonal elements aii= aii-, a is real.Thus every diagonal element of a Hermitian matrix must be real.
• Skew-Hermitian matrix : A square matrix A=(aij) is said to be skew-hermitian if aij=-aji for all i and j. The diagonal elements must be either purely imaginary or must be zero._
LECTURE-8Note:1.The diagonal elements of a Hermitian matrix are real2.The diagonal elements of a Skew-hermitian matrix are eigther zero or purely imaginary3. If A is Hermitian(skew-hermitian) then iA is Skew-hermitian(hemitian).4. For any complex square matrix A , AAө is Hermitian5. If A is Hermitian matrix and its eigen values are real
LECTURE-9Unitary matrix: A complex square matrix A=[aij] is said to be unitary if AAө = AөA = I 0 i 0 − i 0 i − i 0 A i 0 ө − i 0 A= = A = ∴ 0 i 0 i 1 0 − i 0 − i 0 0 1 I AAө = = = ∴ A is a unitary matrix Note: 1. The determinant of an unitary matrix has unit modulus. 2. The eigen values of a unitary matrix are of unit modulus.
LECTURE-10Theorem 1: The values of a hermitian matrix are realTheorem 2: The eigen values of a real symmetric matrix are real