Symmetric Matrices, Spectral Theorem, Diagonalization of a Matrix. Quadratic Forms, Definitions of Positive definite, negative definite, positive semi-definite, negative semi-definite, indefinite forms of matrices, Tests for Positive Definiteness, Positive Definite Matrices and Least Squares, Semi Definite Matrices, Singular Value Decomposition, Applications
Class Portions to be covered
No. 55 Symmetric Matrices, Spectral Theorem, 56-57 Diagonalization of a Matrix, Powers and Products of Matrices 58-60 Quadratic Forms Definitions of positive definite, negative definite, positive semi-definite, negative semi-definite, Indefinite forms of Matrices 61 Matlab Class Number 8, 9 – Projections by Least Squares, Eigen Values and Eigen Vectors 62 Tests for Positive Definiteness 63-65 Problems on Positive Definite Matrices, Semi definite Matrices and Least Squares 66-68 The Singular Value Decomposition of a Matrix. Examples 69-70 Pseudo inverse, Mean & Covariance of a Matrix 71 Matlab Class Number 10-SVD and Pseudo-inverse 72 Applications & Revision
Classwork problems:
1. (i)Factor A = ( 3 1 ) into 𝑆⋀𝑆 −1 and hence compute A49 .
2 2 (ii)Find the matrix A whose eigen values are 1, 3 and eigen vectors are 1 1 ( ) 𝑎𝑛𝑑 ( ). -1 1 -1 1 2 1 Answer : (i) Eigen values are 1, 4: 𝑥1 = ( ) x2 = ( ) (ii) ( ). 2 1 1 2 2. 0 2 2 Find all eigen values and eigen vectors of A = (2 0 2) and write 2 2 0 two diagonalising matrices S. Answer : Eigen values are -2, 4. 3. Orthogonally diagonalize the following symmetric matrices as 𝐴 = 𝑆⋀𝑆 −1 = 𝑄⋀𝑄−1 = 𝑄⋀𝑄𝑇 where Q is an orthogonal matrix. 0 -1 -1 2 1 1 (i) (-1 0 -1) (ii) ( 1 2 1) . -1 -1 0 1 1 2 Answer : Eigen values are (i)-2, 1 (ii) 1,4. 4. Write the symmetric matrix which corresponds to the following quadratic forms: (𝑖)𝑄(𝑥) = −2𝑥1 2 − 7𝑥2 2 + 3𝑥3 2 − 2𝑥2 𝑥3 + 4𝑥1 𝑥3 (𝑖𝑖)𝑄(𝑥) = 3𝑥1 2 + 5𝑥2 2 − 6𝑥3 2 − 2𝑥1 𝑥2 + 3𝑥1 𝑥3 − 𝑥2 𝑥3 (𝑖𝑖𝑖)𝑄(𝑥) = 5𝑥 2 − 4𝑥𝑦 + 3𝑦 2 5. 5 2 0 T Compute the quadratic form x Ax for A=( 2 2 -2 ) and 0 -2 3 𝑥1 1 1/2 (a) 𝑥 = (𝑥2 ) (b) 𝑥 = (-2 ) (c) 𝑥 = (1/√2) 𝑥3 3 1/2 6. Decide for or against the positive definiteness of these matrices and write the corresponding quadratic form f = xTAx. 3 1 0 0 0 0 1 0 -2 3 5 5 -2 3 5 3 ( ) ( ) ( ) (1 3 0) (0 1 2). ( 0 2 -2 ) 5 5 2 3 3 4 0 0 3 0 2 1 -2 -2 7 7. For which a does the matrix A have all 𝜆 > 0 and is therefore positive 𝑎 1 1 definite. 𝐴 = (1 a 1) 1 1 a Answer : For a > 1, A is positive definite. 8. With positive pivots in D. the factorization A=LDLT becomes 𝐴 = 𝐿√𝐷√𝐷𝐿𝑇 , If 𝑅 = 𝐿√𝐷 gives A=RRT (=CCT Cholesky factorization), 2 0 3 6 then from 𝑅 = ( ) find A and from 𝐴 = ( )find R. 1 3 6 16 4 2 Answer : 𝐴 = ( ); 𝑅 = (√3 0 ). 2 10 2√3 2 9. Write A=(3 2 ) as RTR in three ways (𝐿√𝐷)(√𝐷𝐿𝑇 ),(𝑄√𝛬)(√𝛬𝑄𝑇 ) and 1 2 (𝑄√𝛬𝑄 )(𝑄√𝛬𝑄𝑇 ) using pivots eigen values and eigen vectors. 𝑇
10. Compute AAT and ATA and their eigen values and unit eigen vectors, for 7 1 A=(0 0) 5 5 11. Find SVD of the following matrices: 1 1 2 -1 4 2 0 ( ) ( 0 1) ( ) 2 2 -2 -1 0 -1 1 12. Convert the matrix of observations to mean deviation form and construct the sample covariance matrix for the following data: 19 22 6 3 2 20 12 6 9 15 13 5
