On Lei, Miranda, and Thompson's result on singular values and diagonal elements (Q1379106)

Let \(A_1, \dots, A_m\) be real \(n\times n\) matrices with prescribed singular values \(s_1(A_j) \geq\cdots \geq s_n(A_j)\), \(j=1, \dots,m\). Write \(s_i= \prod^m_{j=1} s_i(A_j)\), \(i=1, \dots,n\). The author shows (Theorem 4) that the locus of the diagonal vector \((x_{11}, \dots, x_{nn})\) of the product \((x_{ij}) =U_1A_1 \dots U_mA_m U_{m+1}\) as \(U_1, \dots, U_{m +1}\) run independently over \(SO(n)\) is the convex hull of the set points \((\pm s_{\sigma (1)}, \dots, \pm s_{\sigma (n)})\), where \(\sigma\in \text{Sym} (n)\) and the number of minus signs is even (odd) according as \(\text{det} (A_1 \dots A_m) \geq 0(\leq 0)\). He also determines (Theorem 6) the range of values of certain numerical functions of \(x_{11}, \dots, x_{nn}\). His results generalize various results of \textit{H. Miranda} and \textit{R. C. Thompson} [Linear Algebra Appl. 248, 61-66 (1996; Zbl 0880.15026)] and \textit{T.-G. Lei} [Linear Multilinear Algebra 40, No. 1, 47-59 (1995; Zbl 0844.15012)] for the cases where the \(A_i\) are complex (or real) and the \(U_j\) run over \(U(n)\) (or \(O(n))\).