Do similar matrices have equal singular values?

Question

Is it true that if $A$ and $B$ are similar matrices, $B=S^{-1}AS$, then $A$ and $B$ have the same singular values?

Answer 1

Clearly not. E.g. $\left\{A(x)=\pmatrix{0&x\\ 0&0}:\ x\ne0\right\}$ is a family of similar matrices, but the singular values of $A(x)$ are $|x|$ and zero.

Answer 2

There are plenty of counterexamples and they need not be pathological in any way. Here is another class of counterexamples that is easy to understand.

The matrices whose singular values are all $1$, are the orthogonal matrices. But it is not true that for a (say) diagonal matrix $a$ and orthogonal $k$, the matrix $ak a^{-1}$ is again orthogonal. In fact, this is almost never the case and for it to happen, you need either $k = 1$ or $a$ must have repeated diagonal entries.

So very concretely, take $A$ to be any nontrivial orthogonal matrix and $B = S^{-1} A S$ with $S$ a diagonal matrix with distinct positive diagonal entries.