INEQUALITIES INVOLVING DETERMINANTS
INEQUALITIES INVOLVING DETERMINANTS
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DOI：
10.1090/trans2/032/11
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发表时间：
1955
期刊：
影响因子：
--
通讯作者：
H. Loo
中科院分区：
文献类型：
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作者：
H. Loo
在多元函数论的研究中，我们发现了如下的纯代数不等式：若I-ZZ~′0和I-WW~′0，则d（I-ZZ~′）d（I-WW~′）≤| d（I-ZW~′）|~2. (1)用大写拉丁字母表示n阶复矩阵，用Z~'表示Z的转置共轭复矩阵。如果H是Hermite的，我们用H 0表示H是正定的，用H≥0表示H是半正定的。我们也用d（Z）表示Z的行列式。由于（I-ZW <$'）（I-WW <$'）~（-1）（I-ZW <$'）~' -（I-ZZ <$'）=（Z-W）（I-W~' W）~（-1）（Z-W）<$'，我们推出：|d（I-ZW '）|~2≥d（I-ZZ“）d（I-WW”）-|d（Z-W）|~2，因此，我们有（1）。本文还将该不等式推广到更一般的形式：设X_1，…，X_m为m阶n阶矩阵。设ρ为正数。若I-X_iX_i ∈ '0，其中1 ≤ i ≤ m，则Hermite矩阵是半正定的，这个结果的证明与（1）的证明不同，它需要用到与线性群表示理论有关的引理。找到它的纯代数证明似乎很有趣。
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 H0 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.