A Pythagorean theorem for partitioned matrices (Q6602140)

The paper extends the Pythagorean theorem to matrices partitioned into compatible blocks, either row- or column-wise. The theorem, which involves isometries, is stated in terms of the absolute values, i.e., the positive semi-definite parts of the blocks of the partitioned matrix. For a matrix \(A\), the positive definite part \(|A|\) defined via its polar decomposition \(A=U|A|\), i.e., \(|A| = (A^*A)^{1/2}\). A key step in proving the theorem is the following matrix decomposition (2). It is equivalent to (1) which appears as Lemma 3.4 in the authors' paper [Bull. Lond. Math. Soc. 44, No. 6, 1085--1102 (2012; Zbl 1255.15028)].\N\NDenote by \(\mathbb M_{n+m}\) the set of all \((n+m)\times (n+m)\) complex matrices. Given a positive semidefinite matrix in \(\mathbb M_{n+m}\) partitioned as\N\[\N\begin{bmatrix} A & X \\\NX^* & B \end{bmatrix},\N\]\Nwith diagonal blocks \(A \in \mathbb M_n\) and \(B \in \mathbb M_m\), there exist two unitary matrices \(U, V \in \mathbb M_{n+m}\) such that \N\[\N\begin{bmatrix} A & X \\\NX^* & B \end{bmatrix} = U \begin{bmatrix} A & 0 \\\N0 & 0 \end{bmatrix} U^* + V \begin{bmatrix} 0 & 0 \\\N0 & B \end{bmatrix} V^*, \N\]\Nor, equivalently, \N\[\N\begin{bmatrix} A & X \\\NX^* & B \end{bmatrix} = U_1 A U_1^* + U_2 B U_2^*,\N\]\Nfor two isometry matrices \(U_1 \in \mathbb M_{n+m, n}\) and \(U_2 \in \mathbb M_{n+m, m}\).\N\NThe authors apply this result to derive a theorem by \textit{R. Bhatia} and \textit{F. Kittaneh} [Math. Ann. 287, No. 4, 719--726 (1990; Zbl 0688.47005)]\N for the Schatten \(p\)-norms and provides an inequality for the singular values of compressions onto hyperplanes.