关于矩阵迹的Bellman问题

ON THE BELLMAN'S PROBLEM ABOVT THE TRACE OF MATRICES

设A、B为n阶正定矩阵,所谓Bcllman问题,就是对任何自然数m,猜测不等式Tr(AB)~m≤Tr(A^mB^m)成立。[1]中证明m=2时成立,[2]中证明m=2~k(k为自然数)时成立。本文对m=2的情形作出改进,并证明m=3时猜测成立,给出了等号成立的充要条件。

Let A, B be positive matrices of n×n, the so—called Bellman's problem is to conjecture that for any natural number m, the inquality Tr(AB)~m≤Tr(A^mB^m) is hold. This was shown correct for m=2in [1] and m=2~k (k be natural number) in [2], respectively. In this paper,a revised result was given for m=2, furthermore, the conjecture was shown correct for m=3, and the necessary and sufficient condition for the equality was given.