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michelle34

Guest

linear algebra proof

Hi everyone, any help on this would be greatly appreciated. Thanks!

A is mxm matrix, suppose A^2 = 0. Prove rank(A) <= m/2

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: linear algebra proof

If you use Sylvester’s rank inequality, you will get the answer right away.
If A is a m-by-n matrix and B n-by-k, then
Rank(A)+Rank(B)-n <= Rank(AB).  Which is just 2Rank(A)-m<= Rank(A^2) =Rank(0)
2Rank(A) <=m/2