一個關於行列式的不等式

<正> 在研究多複變數函數論的時候,我們發現了以下的不等式:本文的目的在於給這不等式以一個代數證明,並且把它更精密化些.關於(1)式中所涉及的符號,作以下的說明:在本文中一切拉丁大寫字母都代表n行列的

INEQUALITIES INVOLVING DETERMINANTS

In the study of the theory of functions of several complex variables, we discovered the following pure algebraic inequality:If I-ZZ~′>0 and I-WW~′>0, then d(I-zz~′) d(I-WW~′)≤| d(I-ZW~′)|~2.(1) We use capital Latin letters to denote n-rowed matrices with complex elements, and use Z~' to denote the transposed and conjugate complex matrix of Z. If H is Hermitian, we use H>0 to denote that H is positive definite and H≥0 to denote that H is positive semi-definite. We use also d(Z) to denote the determinant of Z. Since (I-ZW~') (I-WW~')~(-1) (I-ZW~')~' - (I-ZZ~') = (Z-W) (I-W~' W)~(-1) (Z-W)~', we deduce that |d(I-ZW~') |~2≥d(I-ZZ~') d(I-WW~')- |d(Z-W)|~2, and consequently, we have (1). The inequality is also generalized to the following more general form:Let X_1, …, X_m be m n-rowed matrix. Let ρ be a positive number. If I-X_i X_i~'>0 for 1 ≤ i ≤ m, then the Hermitian matrix is positive semi-definite.The proof of this result is different from that of (1), it requires some lemmas related to the representation theory of linear group. It seems to be interesting to find a pure algebraic proof of it.