# What linear transformations are singular values and/or condition numbers invariant under?

This is a purely academic question. Given the singular value decomposition of a matrix, $A=USV^T$, what is the largest set of linear transformations which leave the singular values invariant?

I can see right away that any left or right matrix multiplication by an orthogonal matrix will do the trick. The reason is that orthogonal matrices are closed under matrix multiplication. Therefore, if $O$ is an orthogonal matrix with the dimensions of $U$, then $OA$ has an decomposition of $OA=\tilde{U}SV^T$ where $\tilde{U}\equiv OU$. Therefore, the singular values of $A$ and $OA$ are obviously the same. A similar argument holds for right multiplication. However, are orthogonal transformation the only transformations that will leave the singular values unchanged?

Also, which linear transformations leave the condition number invariant? I would assume this is a much larger set of transformations than those that satisfy the first question, but I'm not sure how to characterize it.

• Probably the answer is that only left or right multiplication by orthogonal matrices will work, although I haven't checked. Jun 24, 2017 at 7:06