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I have a n-by-k matrix A and a n-by-n matrix B, where B is positive definite. I can form the matrix $M = A^t B A$. Playing around, I always found $rk(M) = rk(A)$ but I can't prove this.

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Hint: M and A have the same null space. – Mark Meckes Mar 31 '10 at 12:26
Thanks, I actually found that, but missed that A and M have the same number of columns to finish the proof. – Frank Meulenaar Mar 31 '10 at 12:28
up vote 4 down vote accepted

$A^T BAx = 0 \implies (Ax)^TB(Ax)=0 \implies Ax=0$ by positve definiteness of $B$. So $ker(M)=ker(A)$ and hence $rk(M)=rk(A)$.

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This is rather tangential, but I can't help mentioning that this is related to the fact that in a hilbert space, for a (compact) linear operator one has: $Ker A^* \perp Im A$.

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