I'm supposed to prove this step as part of my proof for existence of Cholesky Decomposition. I can see how to use it in my proof, but I can't seem to be able to prove this lemma:(adsbygoogle = window.adsbygoogle || []).push({});

For any positive (nxn) matrix [tex]A[/tex] and any non-singular (nxn) matrix [tex]X[/tex], prove that

[tex]B=X^{\dagger}A X[/tex]

is positive.

____

Let [tex]X=\left(x_{1}, x_{2}, \ldots, x_{n}\right)[/tex], where all xi are n-vectors.

I see that

[tex] b_{i,j}=x_{i}^{\dagger}Ax_{j}[/tex],

and thus all of the diagonal elements of B are positive (from the definition of a positive matrix).

But where do I go from there?

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# Cholesky Decomposition

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