A Pythagorean theorem for partitioned matrices

Abstract:

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities. For instance, if the matrix $\mathbb {A}$ is partitioned into three blocks $A,B,C$, then \begin{gather*} |\mathbb {A}|^3 \ge U|A|^3U^* + V|B|^3V^*+ W|C|^3W^*,\\ \sqrt {3} |\mathbb {A}| \ge U|A|U^* + V|B|V^*+ W|C|W^*, \end{gather*} for some isometries $U,V,W$, and \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where $\mu _j$ stands for the $j$-th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten $p$-norms and to give a singular value version of Cauchy’s Interlacing Theorem.