设A是n阶的矩阵,证明:n

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