1. Eigen Decomposition and Singular Value Decomposition
Based on the slides by Mani Thomas
Modified and extended by Longin Jan Latecki

2. 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

3. A(x) = (Ax) = (x) = (x)
What are eigenvalues?
Given a matrix, A, x is the eigenvector and  is the corresponding eigenvalue if Ax = x
A must be square and the determinant of A -  I must be equal to zero
Ax - x = 0 ! (A - I) x = 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)

4. 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

5. 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,

6. Physical interpretation
Consider a covariance matrix, A, i.e., A = 1/n S ST for some S
Error ellipse with the major axis as the larger eigenvalue and the minor axis as the smaller eigenvalue

7. Physical interpretation
Original Variable A
Original Variable B
PC 1
PC 2
Orthogonal directions of greatest variance in data
Projections along PC1 (Principal Component) discriminate the data most along any one axis

8. 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. Strang’s lectures

9. Multivariate Gaussian

10. Bivariate Gaussian

11. Spherical, diagonal, full covariance

12. Eigen/diagonal Decomposition
Let be a square matrix with m linearly independent eigenvectors (a “non-defective” matrix)
Theorem: Exists an eigen decomposition
(cf. matrix diagonalization theorem)
Columns of U are eigenvectors of S
Diagonal elements of are eigenvalues of
Unique for distinct eigen-values
diagonal

13. Diagonal decomposition: why/how
Let U have the eigenvectors as columns:
Then, SU can be written
Thus SU=U, or U–1SU=
And S=UU–1.

14. Diagonal decomposition - example
Recall
The eigenvectors and form
Recall
UU–1 =1.
Inverting, we have
Then, S=UU–1 =

15. Example continued  Let’s divide U (and multiply U–1) by Then, S= Q
(Q-1= QT )
Why? Stay tuned …

16. Symmetric Eigen Decomposition
If is a symmetric matrix:
Theorem: Exists a (unique) eigen decomposition
where Q is orthogonal:
Q-1= QT
Columns of Q are normalized eigenvectors
Columns are orthogonal.
(everything is real)

17. Spectral Decomposition theorem
If A is a symmetric and positive definite k £ k matrix (xTAx > 0) with i (i > 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

18. Example for spectral decomposition
Let A be a symmetric, positive definite matrix
The eigenvectors for the corresponding eigenvalues are
Consequently,

19. 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

20. 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

21. Example for SVD Let A be a symmetric, positive definite matrix
U can be computed as
V can be computed as

22. Example for SVD
Taking 21=12 and 22=10, 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

23. 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
-1=diag(1/1, 1/1,, 1/n)
If A is singular, then A-1 = (UVT)-1¼ V0-1UT where
0-1=diag(1/1, 1/2,, 1/i,0,0,,0)
Least squares solutions of a m£n system
Ax=b (A is m£n, m¸n) =(ATA)x=ATb ) x=(ATA)-1 ATb=A+b
If ATA is singular, x=A+b¼ (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

24. Applications of SVD in Linear Algebra
Homogeneous equations, Ax = 0
Minimum-norm solution is x=0 (trivial solution)
Impose a constraint,
“Constrained” optimization problem
Special Case
If rank(A)=n-1 (m ¸ n-1, n=0) then x= vn ( is a constant)
Genera Case
If rank(A)=n-k (m ¸ n-k, n-k+1== n=0) then x=1vn-k+1++kvn with 21++2n=1
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

25. 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

26. Singular Value Decomposition
Illustration of SVD dimensions and sparseness

27. SVD example Let Thus m=3, n=2. Its SVD is
Typically, the singular values arranged in decreasing order.

28. Low-rank Approximation
SVD can be used to compute optimal low-rank approximations.
Approximation problem: Find Ak of rank k such that
Ak and X are both mn matrices.
Typically, want k << r.
Frobenius norm

29. Low-rank Approximation
Solution via SVD
set smallest r-k
singular values to zero k
column notation: sum
of rank 1 matrices

30. Approximation error How good (bad) is this approximation?
It’s the best possible, measured by the Frobenius norm of the error:
where the i are ordered such that i  i+1.
Suggests why Frobenius error drops as k increased.