设A是n阶矩阵,若存在正整数k,使线性方程组A^kα=0有解向量，且A^(k-1)α≠0

A^(k+1)α= A(A^kα) = A0 = 0 其余类似 A^(k+i) = A^i A^kα = A^i0 = 0. 若 A^(k-i)α=0, i>=2 则 A^(k-1)α = A^(i-1) A^(k-i)α = A^(i-1) 0 = 0. 与已知矛盾