How to prove if two matrices are orthogonally equivalent, then they have the same singular values

Question

The question was, "Show that if two matrices are orthogonally equivalent, then they have the same singular values, and there are simple relationships between their singular vectors"

I tried to show like this.

Let $ =^∗ $ for some unitary $ $,

Suppose $ =_ Σ_ _^∗ $ $ =_ Σ_ _^∗ $

$ _ Σ_ _^∗=_ Σ_ _^∗ ^∗ $

$ Σ_ = _^∗ (_ Σ_ _^∗ ^∗)_ $

$ Σ_ ^∗=(_^∗ _) Σ_ (_^∗ ^∗ _) $

Any tips to prove this ?

Answer 1

Let $\mu $ a singular value of $A$. Then there is $x \ne 0$ such that

$A^{\star}Ax= \mu x$. Let $y=Q^{\star}x$. Then $y\ne 0$ and

$ \mu x=QB^{\star}Q^{\star}QBQ^{\star}x=QB^{\star}BQ^{\star}x$ thus

$$ \mu y=B^{\star}By$$

and $\mu$ is a singular value of $B$