关于张量空间上的非负定算子

ON THE NON-NEGATIVE DEFINITE OPERATOR IN THE TENSOR PRODUCT SPACE

设A_1,…,A_n为n阶复矩阵,=A_1…A_n,令W^()={(x_1…x_n,x_1…x_n)|x_1,…,x_n规格化正交}。本文证明了当n≥3时有:1)为非负定的充要条件是W^()R^+;2)为正定的充要条件是W^()R^+(正实数)。

Let A_1,…,A_n be n-square complex matrices,=A_1…A_n.let W^() ={(x_1…x_n,x_1…x_n)|x_1,…,x_n orthonormal}.This note proves that if n≥3, then (a) is non-negative definite iff W^()R^+; (B) is positive definite iff W^()R^+ (positive real).