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Let $A$ be a $p\times p$ positive semi definite matrix.

(1) If $A$ is positive definite, and $B$ is a $p\times q$ matrix with rank $p\leq q$. Show that $B^TAB$ is positive definite.

(2) If $rankA=s\leq p$. Show that there exists a $p\times s$ matrix $M$ such that $A=MM^T$ and $M^TM$ is a diagonal matrix of the positive eigenvalues of A.

I don't even know how to start. Can someone give some hints?

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Some hints:

For part (1), consider the fact that $A$ being positive definite means $$\mathbf{x}^\mathrm{T}A\mathbf{x} > 0$$ for all $\mathbf{x}$. Can you conclude something similar with $B^\mathrm{T}AB$?

For part (2), I would orthogonally diagonalize the matrix. Split the diagonal matrix into it's square root. Are the eigenvectors corresponding to the zero eigenvalues needed? Doing an numerical example will probably show you how to proceed.

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  • $\begingroup$ I get some idea. I didn't think about orthogonally diagonalizing the matrix. Thank you! $\endgroup$ – Frank Lu Oct 19 '12 at 2:25
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(1) I think what you mean is that $B^TAB$ is positive semi-definite, which is easy to show if notice that $$B^TAB=B^TA^{1/2}A^{1/2}B=(A^{1/2}B)^TA^{1/2}B,$$ where $A^{1/2}$ is the square root of $A$, so ${A^{1/2}}^T=A^{1/2}$.

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