Under which operations are singular values invariant?

Question

Take a matrix A and construct B by a permutation of the columns of A. Apparently A and B have the same singular values, but I can't prove it for non 2x2 matrices.

Are there any other operations on matrices that preserve the singular values?

Answer 1

Multiplying $A$ by an orthogonal matrix will necessarily preserve its singular values.

Here's an argument that proves this for square matrices. The singular values of $A$ are equal to the square roots eigenvalues of $A^TA$, which are equal to the square roots of the eigenvalues of $AA^T$. If we multiply $A$ by the left by an orthogonal matrix $U$, then we find that $$ (UA)^T(UA) = A^TU^TUA = A^TA. $$ Thus, the singular values of $UA$ are the square roots of the eigenvalues of $A^TA$, which are the singular values of $A$. Similarly, if we multiply $A$ from the right by an orthogonal matrix $U$, we have $$ (AU)(AU)^T = AUU^TA^T = AA^T. $$ So, the singular values of $AU$ are the square roots of the eigenvalues of $AA^T$, which are the singular values of $A$.

If $U$ is a permutation matrix, then $AU$ is a matrix constructed by permuting the columns of $A$ and $UA$ is a matrix constructed by permuting the rows of $A$.