酉矩阵的子式与余子式的关系

RELATION BETWEEN MINORS AND COFACTORS OF AN UNITARY MATRIX

一、引言一般矩阵的子式与它的余子式是毫无关系的,但酉矩阵(或实正交矩阵)则不然.本文导出它们之间的关系.特别有趣的是,n 阶酉矩阵的任何 m 阶子式与它的 n-m 阶余子式其绝对值相等.

Suppose U is an n×n unitary matrix.For 1≤m≤n,let Q_(m,n) denote the totality of strictly increasing sequences of m positive integers chosen from 1,… n.For α,β∈Q_(m,n),let U[α|β]denote an m×m submatrix of U lying in rows α and columns β,U(α|β)an(n-m)×(n-m)submatrix of U excluding rows α and columnsβ.This paper proves an interesting result: detUdet=(-1)^(s(α)+s(β))det U(α|β) where s(α)=α(1)+…+α(m).In particular,|det U[α|β]|=|det U(α|β)| and detU=detU(α|α)/det if detU[α|α]≠0. This paper also gives a symbolic proof about Cauchy-Binet Theorem and Laplace Expansion Theorem of determinant.