厄米特矩阵的迹的几点性质

应用矩阵Ａ＝（ａｉｊ）∈Ｃｎ×ｎ的弗罗伯尼范数ＡＦ和谱范数ＡＳ，研究厄米特矩阵的迹的性质，得到几个结论：Ｔｒ（ＡＢ）＝∑ｎｉ＝１λｉ∑ｎｊ＝１ｔｉｊμｊ（λｉ，μｊ分别为Ａ，Ｂ的特征值，０≤ｔｉｊ≤１，且∑ｎｉ＝１ｔｉｊ＝１，ｊ＝１，２，…，ｎ）；Ｔｒ（ＡＢ）≤Ｔｒ（Ａ）ＢＳ；Ｔｒ（ＡＢ）Ｈ（ＡＢ）］≤Ｔｒ（ＡＨＡ）［ｍａｘ１≤ｉ≤ｎλｉ］２（λｉ是Ｂ的特征值）等．

On Several Properties of Trace of Hermitian Matrix

By virtue of Frobenius norm ‖A‖ F and Speetral norm ‖A‖ S of matrix A=(a ij )∈C n×n , we make a careful study of the properties of Hermitian matrix's trace, obtaining several conclusions, i.e. Tr( AB)=∑ni=1λ i∑nj=1t ij μ j(λ i,μ j is respectively the eigenvalue of A,B , and t ij is a group of nonnegative real numbers: 0≤t ij ≤1 , and ∑ni=1t ij =1,j=1,2,…,n) ;Tr (AB)≤ Tr (A)‖B‖ S ;Tr[ (AB) H(AB) ]≤Tr (A HA) [max 1≤i≤nλ i] 2(λ i is B′s eigenvalue) and so on.