Lecture 15 Review of Matrix Theory III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Transcription

1 Lecture 15 Review of Matrix Theory III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

2 Matrix An m n matrix is a rectangular or square array of elements with m rows and n columns. Eg: A m n a a a a a a n n = a a a m1 m2 mn For each subscipt, a, i = the row, and j = the column. ij If m = n, then the matrix is said to be a "square matrix". 2

3 Vector If a matrix has just one row, it is called a "row vector" Eg. [ b b b ] n If a matrix has just one column, it is called a "column vector" Eg. c c c m1 3

4 Null Matrix / Diagonal Matrix Null matrix : A matrix with all zero elements Eg. A = Diagonal Matrix : A square matrix with all off diagonal elments being zero a a Eg. A = 0 0 a

5 Linear Dependence/Independence A set of n column vectors s1, s2,, sn, is said to be "linearly dependent" if there exist constants α1, α2,, αn, not all zero, such that α s + α s + + α s = If the above equation holds only when α1= α2 = = α n, then the vectors are "linearly independent". n n 5

6 Example: Linear Dependence/Independence Question : 3 [ ] T [ ] a [ ] Are the vectors a = 1 1 0, = 1 0 1, a = T linearly independent? T Answer : We must test whether xa1 + ya2 + za3 = 0 admits any non-trivial solution. The determinant of the coefficient is D = = Hence the vectors are linearly independent. 6

7 Rank of a Matrix The rank of a matrix A m n is equal to the number of linearly independent rows or columns. The rank can be found by finding the higest-order square submatrix that is non-singular Eg. A = Here A = 0. So, the A matrix is singular and hence rank( A) Choosing the sub-matrix B=, B = Hence rank( A) = 2 7

8 Matrix Operations Addition The sum of two matrices, written A + B = C, is defined by aij + bij = cij Eg = -1 8 Subtraction The difference between two matrices, written A B = C, is defined by aij bij = cij Eg. 3 5 =

9 Matrix Operations Multiplication The product of two matrices, written AB = C, is defined by c = a b a a a b b b Eg. A = ; B b21 b22 b 23 a21 a22 a = 23 b31 b32 b33 AB = n ij ik jk k = 1 ab ab ab ab a b + a b a b + a b + a b a b + a b + a b a b + a b + a b a b + a b + a b Schur / Hadamard / Direct product AB i = aij ib ij, if A& Bhave some dimension 9

10 Matrix Identities Commutative Law Associative Law Transpose of sum Transpose of product A + B = B + A AB BA Α + B + C = A + B + C ( ) ( ) Α BC = ( ) ( ) ( ) ( ) AB C T T T A + B = A + B AB T T T T A = AB AB = B A A A BA Determinant identities det det det = det det det = det B 10

11 Trace Trace of T r Α n n Α = ( ) n i= 1 a ij : Sum of the diagonal elements 1 3 Eg A =, Tr( Α ) = =

12 Example: Eigenvalues/Eigenvectors 1 3 Question : Find eigenvalues/eigenvecotrs of A = 2 2 Solution : Eigenvalues: The characteristic equation is given by 1 λ 3 det ( A λi2 ) = = λ 2 ( λ)( λ) λ λ ( λ )( λ ) 1 2 6= 3 4= = 0 Hence the eigenvalues are λ = 1, λ =

13 Example: Eigenvalues/Eigenvectors Eigenvectors: a1 Let X ( ) 1 =. Then the equation A 1I2 X1 0 leads to: b λ = a1 0 3a1 + 3b1 = 0 ( a ) 1 b b = 1 0 = 2a1 2b1 = 0 The solutions can be expressed as a1 = b1 = α for any α. If we put α = 1, then an eigenvector is 1 X1 = 1 13

14 Example: Eigenvalues/Eigenvectors Similarly ( A λ I ) X = 0 implies a 2a + 3b = 0 = 0, or b + 2 2a2 + 3b2 = 0 The eigenvectors for this case are β X 2 = 2 β 3 for any nonzero β. Assuming β = 3 gives the eigenvector 3 X 2 = 2 14

15 Example: Eigenvalues/Eigenvectors 15

16 Example: Eigenvalues/Eigenvectors 16

17 Example: Eigenvalues/Eigenvectors 17

18 Additional Properties of Eigenvalues/Eigenvectors Α= λ1 λn Tr Α = λ + + λ ( ) If det 1 λi Α = λ + a λ + + aλ + a ( ) n 1 n n ( ) ( ) n n 1 = Α 0 = Α then a Tr, a 1 If Α is Real and Skew-Symmetric n n n ( T Α= Α ) ( Α= Α ) or Complex and Skew-Hermition, Then its eigenvalues are all pure imaginary 18

19 Additional Properties of Eigenvalues/Eigenvectors [ ] If Α n nis Real and Symmetric or Complex and Hermition, then its eigenvalues are all real. If A is a symmetrical matrix, then the eigenvectors associated with two distirct eigenvalues are orthogonal If An nis a symmetric matrix, then T X ΑX λ ( ) ( ) min < R X < λmax,where, R X, X 0 T X X 19

20 Proof 20

21 Proof 21

22 Proof 22

23 Proof 23

24 Generalized Eigenvectors Suppose A has eigenvalues λ, λ, λ Let A A+ εb, ε << 1 ε such that Α has distinct eigenvalues λ, λ, λ ε where λ = λ + δ, δ << and the corresponding eigenvectors ν1, ν2, ν3 = ν2 + δw Then as ε 0, the following equations are satisfied: ( λi ) νi ( A λ2i) w= ν2 { ν ν w} (1) A I = 0 i= 1, 2 (2) Note that,, are linearly independent 1 2 and w is called a "generalited eigenvector" of A. 24

25 Quadratic Forms T = [ ] Suppose X x1, x2,, x n. Then any polynomial function the elements in which every term is of degree two is known as a quadratic form. Thus, if n = 3, then x1 + 8x1x2 + x2 + 6x2x3 + x3 is an example of a quadratic form. Quadratic forms can always be expressed as a matrix product T of the form X AX Utility: Optimization theory, Optimal control theory etc. 25

26 Example: Quadratic Form The example above can be written as x1 x1 x2 x x x [ ] 3 In this representation, A is required to be symmetric matrix, Nonsymmmetric representations are possible with Eg: A = but the symmetric form is "standard" 26

27 Example: Singular Value Decomposition , T A= A A = λ = 4,0,0 1,2,3 ( T AA) X1 = 0, X 2 1, X = = for λ = 4 for λ = 0 for λ = 0 ( ) ( ) ( ) 27

28 Example: Singular Value Decomposition σ1 λ 1 2 σ 2 λ 2 0 = = Note: The values are ordered σ 3 λ Y1 = ( AX1) = 0 σ = [ ] ( ) T Next, find Y = a b such that Y and Y will be orthonormal i.e. Y, Y = Y T T Y2 = a b= 0 and Y2, Y2 = Y2 Y2 = a + b = 1 28

29 Example: Singular Value Decomposition Solution: a b 1/ 2. Hence, Y P 1 2 = = = = 2 = Y Y T X1 X2 X3 1 = = Q σ D = = Verify : PDQ = = = A 29

30 1 Derivative of A ( t) Let Hence ( ) ( ); Then ( ) ( ) 1 A t = B t A t B t = I d dt () () ( At Bt ) () At () db t dt db t dt = 0 () da() t = () () () + Bt = dt da t () 1 A t B t 1 da () t ( ) 1 da t 1 dt dt = A () t A t dt () 0 30

31 Practice Problems Let A=. Find eigenvalues/eigenvectors of. In addition, find 1 2 A the eigenvalues/eigenvectors of A, A m= 3,4 and I + 2A+ 4 A without computing these matrices. 2. Proove ( AB) = B A ( ) 1 m 2,if the indicated matrix inverses exist. 3. An n n Hadamard matrix A has all elements that are ( ) T ± 1 and satisfies A A= ni. Show that det A = n 4. Show that the determinant of a negative definite n n matrix is positive if n is even and negative if n is odd. 5. Let A and P be n n matrices with P nonsingular. Show that 1 ( ) = Tr ( P AP) Tr A n/2 31

32 Practice Problems 6. Following the standard definitions, Show that for a fixed X n R X X as p 1 2 ( ) 7. Compute A, A, A and p A of the matrix in Problem-1 8. Find the singular value decomposition of ( ( )) A = Give all the steps. Verify the results using MATLAB. 1 T 9. Show that X AX = AX X 2 n n 10. Show that if F f X R, X R, F F F then = X p n f X T m n p 32

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