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 $ 𝐴=𝑄𝐵𝑄^∗ $ for some unitary $ 𝑄$,
Suppose $ 𝐴=𝑈_𝐴 Σ_𝐴 𝑉_𝐴^∗ $ 𝑎𝑛𝑑 $ 𝐵=𝑈_𝐵 Σ_𝐵 𝑉_𝐵^∗ $
$ 𝑈_𝐴 Σ_𝐴 𝑉_𝐴^∗=𝑄𝑈_𝐵 Σ_𝐵 𝑉_𝐵^∗ 𝑄^∗ $
$ Σ_𝐴 = 𝑈_𝐴^∗ (𝑄𝑈_𝐵 Σ_𝐵 𝑉_𝐵^∗ 𝑄^∗)𝑉_𝐴 $
$ 𝐼Σ_𝐴 𝐼^∗=(𝑈_𝐴^∗ 𝑄𝑈_𝐵) Σ_𝐵 (𝑉_𝐵^∗ 𝑄^∗ 𝑉_𝐴) $
Any tips to prove this ?