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 $\Lambda$ and $M$ have unimodular diagonal entries. Denote by $D_A$ and $D_B$ a pair of diagonal matrices whose cosquares are $\Lambda$ and $M$ respectively. The congruence orbits generated by $D_A$ and $D_B$ 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, $\tilde{A}= P^*D_AP$, $\tilde{B}= P^*D_BP$. Then $\tilde{A}$ and $\tilde{B}$ satisfy the relations $\tilde{A}\tilde{B}^{-*}=\tilde{B}\tilde{A}^{-*}$ and $\tilde{A}^{-*}\tilde{B}=\tilde{B}^{-*}\tilde{A}$. In theory of congruences, these relations can be regarded as a kind of substitute for the conventional permutability.
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