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Unit 4 Classwork - Docla 24

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UE22MA241B LINEAR ALGEBRA AND ITS APPLICATIONS

Unit 4: Singular Value Decomposition


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

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