A PYTHAGOREAN THEOREM FOR PARTITIONED MATRICES

초록

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 is partitioned into three blocks A, B, C , then |A|(3 )>= U |A|U-3 & lowast; + V|B|V-3 & lowast; + W|C|W-3 & lowast;, root 3|A| >= U|A|U & lowast; + V|B|V & lowast; + W|C|W & lowast;, for some isometries U, V, W, and mu (2)(4)(A) <= mu (2)(3)(A) + mu (2)(2)(B) + mu (2)(1)(C) 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.

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