A criterion for unitary congruence between complex matrices
The following new theorem of unitary congruence of two matrices is proved: Let \(A,B \in {M_n}({\mathbb C})\), where \({M_n}({\mathbb C})\) is the set of complex matrices of order \(n\), and \[ \begin{aligned} &\alpha (s) = A{A^ * },\,\,\alpha (t) = {A^T}\bar A,\,\,\alpha (u) = A\bar A,\,\,\alpha (v) = {(A\bar A)^ * },\\ &\beta (s) = B{B^ * },\,\,\beta (t) = {B^T}\bar B,\,\,\beta (u) = B\bar B, \beta (v) = {(B\bar B)^ * }\end{aligned} \] be the families over the alphabet \(X = \{ s,t,u,v\}\) determined by \(A\) and \(B\). Then \(A\) and \(B\) are unitarily congruent if and only if the conditions \(tr\alpha (p) = tr\beta (p)\), which form a finite set, are fulfilled for every \(p \in {X^{2l + 1}}\) and for a certain upper bound integer \(l = l(n)\) for the length of all families in \({M_n}({\mathbb C})\).
scientific article; zbMATH DE number 6110777
scientific article