SVD for the Vertically Challenged

If you haven't noticed, this blog can be a bit random, and I never know what will garner interest on the Net. One surprisingly popular offering has been the little SVD calculator I created by porting the JAMA SVD code from Java to JavaScript.

Today, I found an obvious bug in my applet and fixed it. In the process, I discovered a number of people querying the Net about the issue leading to the bug. The problem was that the JAMA code only calculated the SVD in cases where m >= n; i.e., it could only calculate the SVD on matrices with more rows (m) than columns (n).

This is easily fixed, if you consider the matrix transpose and corresponding transpose of its SVD.

A^T = (U*S*V^T)^T

The basic rule of matrix algebra to consider: The transpose of a matrix product is equal to the reverse product of the individual matrix transposes (cf. The Matrix Cookbook).

In other words:

(U*S*V^T)^T = V*S^T*U^T

Bottom line: If you can only calculate the SVD of A where m >= n, all you need to do is calculate the SVD of A^T and swap the resulting U and V matrices.