Singular values, diagonal elements, and extreme matrices (Q1968761)

Here are characterizations of the equality cases for two inequalities established earlier by \textit{H. Miranda} and \textit{R. C. Thompson} [ibid. 185, 165-172 (1993; Zbl 0774.15012)]. Let \(A\) and \(B\) be complex matrices with respective singular values \(a_1\geq\cdots\geq a_n\) and \(b_1\geq\cdots\geq b_n\). Let \(d_1,\dots, d_n\) denote the diagonal elements of \(AB\), ordered so that \(|d_1|\geq\cdots\geq|d_n|\). Let \(1\leq k\leq n\). Then \[ |d_1|+\cdots+|d_n|= a_1b_1+\cdots+ a_n b_n \] if and only if there exist unitary \(U\), \(V\), \(W\) such that \[ U^*AV= \text{diag}(a_1,\dots, a_n)\oplus A_2,\quad V^*BW= \text{diag}(b_1,\dots, b_n)\oplus B_2, \] and \(AB= (AB)_1\oplus (AB)_2\) where \((AB)_1\) is \(k\times k\) and for some diagonal \(D\), \(D(AB)_1\) is positive semidefinite with eigenvalues \(a_1b_1+\cdots+ a_kb_k\). A similar decomposition is shown to be equivalent to having \[ |d_1|+\cdots+|d_{n-1}|-|d_n|= a_1b_1+\cdots+ a_{n-1} b_{n-1}- a_n b_n. \]