Question

5. We now have all the vocabulary needed to discuss a very important tool in linear algebra- something called singular value decomposition (SVD). Let M be an m x n matrix. Give a definition of the SVD of M. We have yet to talk about this in class, so you will need to search for the definition on the internet or in a textbook- whatever you do, cite your source. To keep things simple don't worry about the words "complex" or "unitary", meaning your definition should assume M and the matrices in the SVD of M are real (a real matrix has components that are real numbers as opposed to a complex matrix, which has components that are complex numbers).

Answer 1

Singular value decomposition is a process to factorise any real or complex matrix which is a generalization of eigen value decomposition of n×n matrix . As we see it is a generalization so the matrix can be rectangular as well.

Singular Value decomposition of matrix M of m×n is given as M=ULV

where ,

U is m×m real or complex unitary matrix

L is m×n rectangular diagonal matrix with diagonal element non negative real number

V is n×n real or complex unitary matrix

In particular if M is real then U and V are real orthogonal matrices