Page 105 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

3. Numerical Linear Algebra 93

3. SINGULAR VALUE DECOMPOSITION (SVD)

As discussed earlier any real square matrix ‘A’ can be diagonalized using
eigen matrix ‘E’ (every column vector is the eigen vector of the matrix ‘A’)

A=E D E T

where ‘D’ is the diagonal matrix filled with Eigen values in the diagonal

Suppose if the matrix ‘A’ is the rectangular matrix of size mxn, then the
matrix ‘A’ can be represented as the follows

T
A=U ^ V . This is called as Singular Value Decomposition.

T
AA is the square matrix with size mxm. Using Eigen decomposition
T
AA = U D 1 U T
T
Similarly A A of size nxn can be reprersented using Eigen
decomposition as

T
T
A A= V D 2 V

T
If A = U ^ V
T
T
A =V ^ U T
T
Note that ^ and ^ are the same as the matrix is the diagonal matrix.
T
T
T
AA = U ^ V V ^ U T

= U ^^ U [Expected Result]
T
T

Similarly

T
T
T
A A = V ^ U U ^V T

T
T
= V ^ ^ V [Expected Result]
The diagonal elements of the matrix ^ is obtained as the positive square
root of the diagonal elements of the matrix D 1 or D 2. Note that the diagonal
elements of the matrix D 1 and D 2 are same except the addition or deletion of
zeros.
Thus the matrix A is represented as the product of the Eigen matrix
T
T
obtained from the matrix AA , Eigen matrix obtained from the matrix A A