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Eigen Decomposition and Singular Value

Decomposition

- Mani Thomas
- CISC 489/689

Introduction

- Eigenvalue decomposition
- Spectral decomposition theorem
- Physical interpretation of eigenvalue/eigenvectors

- Singular Value Decomposition
- Importance of SVD
- Matrix inversion
- Solution to linear system of equations
- Solution to a homogeneous system of equations
- SVD application

What are eigenvalues?

- Given a matrix, A, x is the eigenvector and ? is

the corresponding eigenvalue if Ax ?x - A must be square the determinant of A - ? I must

be equal to zero - Ax - ?x 0 ! x(A - ?I) 0
- Trivial solution is if x 0
- The non trivial solution occurs when det(A - ?I)

0 - Are eigenvectors are unique?
- If x is an eigenvector, then ?x is also an

eigenvector and ?? is an eigenvalue - A(?x) ?(Ax) ?(?x) ?(?x)

Calculating the Eigenvectors/values

- Expand the det(A - ?I) 0 for a 2 2 matrix
- For a 2 2 matrix, this is a simple quadratic

equation with two solutions (maybe complex) - This characteristic equation can be used to

solve for x

Eigenvalue example

- Consider,
- The corresponding eigenvectors can be computed as
- For ? 0, one possible solution is x (2, -1)
- For ? 5, one possible solution is x (1, 2)

For more information Demos in Linear algebra by

G. Strang, http//web.mit.edu/18.06/www/

Physical interpretation

- Consider a correlation matrix, A
- Error ellipse with the major axis as the larger

eigenvalue and the minor axis as the smaller

eigenvalue

Physical interpretation

- Orthogonal directions of greatest variance in

data - Projections along PC1 (Principal Component)

discriminate the data most along any one axis

Physical interpretation

- First principal component is the direction of

greatest variability (covariance) in the data - Second is the next orthogonal (uncorrelated)

direction of greatest variability - So first remove all the variability along the

first component, and then find the next direction

of greatest variability - And so on
- Thus each eigenvectors provides the directions of

data variances in decreasing order of eigenvalues

For more information See Gram-Schmidt

Orthogonalization in G. Strangs lectures

Spectral Decomposition theorem

- If A is a symmetric and positive definite k k

matrix (xTAx gt 0) with ?i (?i gt 0) and ei, i 1

? k being the k eigenvector and eigenvalue pairs,

then - This is also called the eigen decomposition

theorem - Any symmetric matrix can be reconstructed using

its eigenvalues and eigenvectors - Any similarity to what has been discussed before?

Example for spectral decomposition

- Let A be a symmetric, positive definite matrix
- The eigenvectors for the corresponding

eigenvalues are - Consequently,

Singular Value Decomposition

- If A is a rectangular m k matrix of real

numbers, then there exists an m m orthogonal

matrix U and a k k orthogonal matrix V such

that - ? is an m k matrix where the (i, j)th entry ?i

0, i 1 ? min(m, k) and the other entries are

zero - The positive constants ?i are the singular values

of A - If A has rank r, then there exists r positive

constants ?1, ?2,??r, r orthogonal m 1 unit

vectors u1,u2,?,ur and r orthogonal k 1 unit

vectors v1,v2,?,vr such that - Similar to the spectral decomposition theorem

Singular Value Decomposition (contd.)

- If A is a symmetric and positive definite then
- SVD Eigen decomposition
- EIG(?i) SVD(?i2)
- Here AAT has an eigenvalue-eigenvector pair

(?i2,ui) - Alternatively, the vi are the eigenvectors of ATA

with the same non zero eigenvalue ?i2

Example for SVD

- Let A be a symmetric, positive definite matrix
- U can be computed as
- V can be computed as

Example for SVD

- Taking ?2112 and ?2210, the singular value

decomposition of A is - Thus the U, V and ? are computed by performing

eigen decomposition of AAT and ATA - Any matrix has a singular value decomposition but

only symmetric, positive definite matrices have

an eigen decomposition

Applications of SVD in Linear Algebra

- Inverse of a n n square matrix, A
- If A is non-singular, then A-1 (U?VT)-1

V?-1UT where - ?-1diag(1/?1, 1/?1,?, 1/?n)
- If A is singular, then A-1 (U?VT)-1¼ V?0-1UT

where - ?0-1diag(1/?1, 1/?2,?, 1/?i,0,0,?,0)
- Least squares solutions of a mn system
- Axb (A is mn, mn) (ATA)xATb ) x(ATA)-1

ATbAb - If ATA is singular, xAb¼ (V?0-1UT)b where ?0-1

diag(1/?1, 1/?2,?, 1/?i,0,0,?,0) - Condition of a matrix
- Condition number measures the degree of

singularity of A - Larger the value of ?1/?n, closer A is to being

singular

http//www.cse.unr.edu/bebis/MathMethods/SVD/lect

ure.pdf

Applications of SVD in Linear Algebra

- Homogeneous equations, Ax 0
- Minimum-norm solution is x0 (trivial solution)
- Impose a constraint,
- Constrained optimization problem
- Special Case
- If rank(A)n-1 (m n-1, ?n0) then x? vn (? is

a constant) - Genera Case
- If rank(A)n-k (m n-k, ?n-k1? ?n0) then

x?1vn-k1??kvn with ?21??2n1

- Has appeared before
- Homogeneous solution of a linear system of

equations - Computation of Homogrpahy using DLT
- Estimation of Fundamental matrix

For proof Johnson and Wichern, Applied

Multivariate Statistical Analysis, pg 79

What is the use of SVD?

- SVD can be used to compute optimal low-rank

approximations of arbitrary matrices. - Face recognition
- Represent the face images as eigenfaces and

compute distance between the query face image in

the principal component space - Data mining
- Latent Semantic Indexing for document extraction
- Image compression
- Karhunen Loeve (KL) transform performs the best

image compression - In MPEG, Discrete Cosine Transform (DCT) has the

closest approximation to the KL transform in PSNR

Image Compression using SVD

- The image is stored as a 256264 matrix M with

entries between 0 and 1 - The matrix M has rank 256
- Select r 256 as an approximation to the

original M - As r in increased from 1 all the way to 256 the

reconstruction of M would improve i.e.

approximation error would reduce - Advantage
- To send the matrix M, need to send 256264

67584 numbers - To send an r 36 approximation of M, need to

send 36 36256 36264 18756 numbers - 36 singular values
- 36 left vectors, each having 256 entries
- 36 right vectors, each having 264 entries

Courtesy http//www.uwlax.edu/faculty/will/svd/co

mpression/index.html

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