一类新的正规矩阵及其性质

A new set of normal matrices and their properties

寻找正规矩阵,是矩阵理论研究的重要课题之一.受Hermite矩阵和参考文献[1]的启发,发现适合条件A^(*)=-A^(2)的矩阵是一类正规矩阵.利用正规矩阵,共轭转置矩阵,矩阵的奇异值等概念和理论,证明了这种矩阵可以对角化以及等式(A⊗B)^(*)=(A⊗B)^(2),给出了它的可能特征值的分布及其谱分解,以及等式(A⊕B)^(*)=-(A⊕B)^(2)成立的充要条件,还给出了这种矩阵的奇异值分解式等.这些结论的获得都用到了共轭转置矩阵,也丰富了共轭转置矩阵的理论.

Searching for normal matrix is one of the important objects in matrix theory.Inspired by Hermitian matrix and article[1],the matrix suited to A^(*)=-A^(3)is found and proved to be those ones.Utilizing the concepts and theories of normal matrix,conjugate transpose matrix,and singular value of matrix and so on,the distribution of the possible characteristic values of the matrix suited to A^(*)=-A3 is given,the formula of(A⊗B)^(*)=(A⊗B)^(3)and the sufficient and necessary condition of(A⊗B)^(*)=-(A⊗B)^(3)are investigated and the singular value decomposition of A above is also obtained.These conclusions will enrich the theory of conjugate transpose matrix.