Abstract
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.
