Simultaneous Diagonalization of Two Matrices: Simularities and Congruences

We prove the following assertion. Let A and B be nonsingular unitoids with simple canonical angles. Assume that their cosquares $${{\mathcal{C}}_{A}}$$ = A–*A and $${{\mathcal{C}}_{B}}$$ = B–*B commute. Bring both cosquares to diagonal form by one and the same similarity (simultaneous diagonalization). The resulting diagonal matrices Λ and M have unimodular diagonal entries. Denote by DA and DB a pair of diagonal matrices whose cosquares are Λ and M, respectively. The congruence orbits generated by DA and DB will be denoted by $$\mathcal{O}({{D}_{A}})$$ and $$\mathcal{O}({{D}_{B}})$$ . Let $$\tilde {A}$$ and $$\tilde {B}$$ be the points of these orbits corresponding to the same transition matrix P, that is, $$\tilde {A}$$ = P*DAP, $$\tilde {B}$$ = P*DBP. Then $$\tilde {A}$$ and $$\tilde {B}$$ satisfy the relations $$\tilde {A}{{\tilde {B}}^{ - }}\text{*}$$ = $$\tilde {B}{{\tilde {A}}^{ - }}\text{*}$$ and $${{\tilde {A}}^{ - }}\text{*}{\kern 1pt} \tilde {B}$$ = $${{\tilde {B}}^{ - }}\text{*}{\kern 1pt} \tilde {A}$$ . In theory of congruences, these relations can be regarded as a kind of substitute for the conventional permutability.