中科新闻网我为祖国庆生日简报:线形代数的问题，高手请回答！谢谢！
来源：百度文库 编辑：中科新闻网 时间：2024/05/06 06:34:24
设A是n阶的矩阵，证明:n<=rank(A+E) + rank(A-E);
Dim(Ker(A+E)) + Rank(A+E) = Dim(A+E) = n
Dim(Ker(A-E)) + Rank(A-E) = Dim(A-E) = n
Rank(A+E) + Rank(A-E)
= 2n - Dim(Ker(A+E)) - Dim(Ker(A-E))
For any V in Ker(A+E), (A+E)V = 0,
so (A-E)V = (A+E)V - 2V = -2V /= 0
V is not in Ker(A-E)
Therefore Dim(Ker(A+E)) + Dim(Ker(A-E)) <= n
Rank(A+E) + Rank(A-E)
= 2n - Dim(Ker(A+E)) - Dim(Ker(A-E))
>= n