# Singular Value Decomposition

1. May 7, 2007

### Mindscrape

I have a couple questions about the singular value decomposition theorem, which states that any mxn matrix A of rank r > 0 can be factored into
$$A = U \Sigma V$$
into the product of an mxm matrix U with orthonormal columns, the mxn matrix ∑ with ∑ = diag($$\sqrt{\lambda_i}$$), and the nxn matrix V with orthonormal columns.

In case the definition doesn't provide much help, the V has the orthonormalized eigenvectors of (A^T)A, and U has the orthonormalized eigenvectors of A(A^T).

Do the first r columns of U span A, i.e. do the first r columns of U form a basis for the range of A? Similarly, will the first r columns of V form a basis for the corng of A?

Really what I am trying to determine is if C is a 3x3 matrix with eigenvalues of 0, 1, and 2, if the eigenvalues of C^T C can be determined with the eigenvalues of C.

2. May 7, 2007

### D H

Staff Emeritus
3. May 7, 2007

### Mindscrape

Actually, I was really certain about the bases for rng and corng.

Mostly what I don't see is the relation between the eigenvalues of C and the eigenvalues of C^T C, because one matrix is a symmetric positive definite matrix, while the other (C) is a general 3x3 matrix.