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 A \mathbb {A} is partitioned into three blocks A , B , C A,B,C , then | A | 3 ≥ U | A | 3 U ∗ + V | B | 3 V ∗ + W | C | 3 W ∗ , 3 | A | ≥ U | A | U ∗ + V | B | V ∗ + W | C | W ∗ , \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 U,V,W , and μ 4 2 ( A ) ≤ μ 3 2 ( A ) + μ 2 2 ( B ) + μ 1 2 ( C ) \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where μ j \mu _j stands for the j j -th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten p p -norms and to give a singular value version of Cauchy’s Interlacing Theorem.