n×n矩阵空间的幂等基与对合基

Idempotent Basis and Involutory Basis of n×n Matrix Space

证明了n≥2时n×n阶矩阵空间存在无穷多个由幂等矩阵构成的幂等基,且对每个n×n阶矩阵给出了标准幂等基显示的线性表出系数.提出n×n阶矩阵空间基的基秩定义,证明了所有幂等基的基秩的最大下界和最小上界都是可达的.同时,首次得到n×n阶矩阵空间的对合基.

When n≥2,there exists not only a basis of n ×n matrix space composed of idempotent matrices,but also infinitely many different idempotent bases.This leads to the conclusion that each n×n matrix can be uniquely linearly represented by idempotent matrices,and the linear expression coefficients are shown. It also gives the definition of base rank of basis of n×n matrix,and proves that the maximum lower bound and the minimum upper bound of the base rank of all idempotent basis are reachable.The involution base of n×n matrix space are obtained firstly.