一类由球带函数诱导的矩阵函数的MPW问题（Ⅲ）

An MPW Problem of the Matrix Functions Induced by Zonal Spherical Function Ⅲ

若矩阵函数ｄ满足对任何－ｎ阶可逆矩阵Ａ，ｄ（Ａ）ｄｅｔ（Ａ－１）＝ｄ（Ａ－１）ｄｅｔ（Ａ），则称ｄ属于类ＭＰＷ。若ｄ满足对任－ｎ阶可逆矩阵Ａ，当且仅当ｄ（Ａ－１）＝０时，ｄ（Ａ）＝０，则称ｄ属于类２．讨论了由球带函数导出的一类矩阵函数ｄ的性质，证得ｄ属于类２的必要充分条件是ｄ属于类ＭＰＷ。

For any invertible matrix A,of the nth order,if a matrix function d satisfies d(A)×det(A-1)=d(A-1) det(A),then d is said to be in class MPW. For any invertible matrixA of the nth order, of d satisfies d d(A)=0 if and only if d(A-1)=0 , then d is said to be inclass 2. The properties of the generalized matrix function d induced by the zonal sphericalfunctions is discussed. It is proved that the necessary and sufficient condition for the general-ized matrix function d to be in class 2 is that d should be in MPW.