Let A = UP be a polar decomposition of an n x n complex matrix A. Then for every unitarily invariant norm ., it is shown that UP - PU(2)less than or equal toA*A - AA* less than or equal to UP + PU UP - PU, where . denotes the operator norm. This is a quantitative version of the well-known result that A is normal if and only if UP = PU. Related inequalities involving self-commutators are also obtained. [References: 12]
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机译：令A = UP为nxn复矩阵A的极坐标分解。然后对于每个unit不变范数。，表明 UP-PU （2）小于或等于 A * A-AA * 小于或等于 UP + PU UP-PU ，其中。表示运营商规范。这是众所周知的结果的定量形式，当且仅当UP = PU时，A才是正常的。还获得了涉及自换向器的相关不等式。 [参考：12]
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