What is the missing piece in proving A=0 when A(A*)=0?

[mathboy]

Let A be a nxn matrix. Prove that if (A*)A=0 then A=0. What if A(A*) = 0?

A* is the conjugate transpose of A. When I write out the expansion formula, I cannot conclude that every entry of A is zero. What am I missing?

 

[Dick]

Concentrate on the diagonal elements of (A*)A. Each one is the inner product of a row of A* with the corresponding column of A. I.e. it's the inner product of a vector with it's conjugate transpose. Under what conditions can that be zero?

 

[mathboy]

Oh, that was a clever idea. I got it now. So it turns out the weaker condition of (A*)A having zero diagaonal entries is enough to conclude that A=0. And the same is true of A(A*).

 

[Dick]

Bingo. You've got it.