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I am having trouble solving a property that I found.

If $A:n \times n$ is defined as a positive definite matrix and $B: n \times m$ where $rank(B) = r$.

Then $B^T A B > 0$, only when r = m and $B^T A B \ge 0$, when $r < m$.

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1 Answer 1

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$B^TAB>0$ iff $(Bx)^TA(Bx)=x^TB^TABx>0$ for all nonzero vectors $x\in\mathbb R^m$. Since $A$ is positive definite, the question now becomes whether $Bx$ could be zero for some nonzero vector $x$. This is where the rank of $B$ matters.

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