Specht's criterion for systems of linear mappings

Abstract

W.Specht (1940) proved that two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\mathrm{trace}w(A,A^{\ast })=\mathrm{trace}w(B,B^{\ast })$ for every word $w(x,y)$ in two noncommuting variables. We extend his criterion and its generalizations by N.A.Wiegmann (1961) and N.Jing (2015) to an arbitrary system $\mathcal{A}$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\mathcal{A})$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\tilde{Q}(\mathcal{A})$ the directed graph obtained by enlarging to $Q(\mathcal{A})$ the adjoint linear mappings. We prove that a system $\mathcal{A}$ is transformed by isometries of its spaces to a system $\mathcal{B}$ if and only if the traces of all closed directed walks in $\tilde{Q}(\mathcal{A})$ and $\tilde{Q}(\mathcal{B})$ coincide.

21 pages

References in corpus (2)

• Complexity of matrix problems
• Unitary and Euclidean representations of a quiver