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a.) Prove that every invertible matrix A has a polar decomposition, written $A = QB$, into the product of an orthogonal matrix $Q$ and a positive definite matrix $B>0$. Show that if $detA>0$, then $Q$ is a proper orthogonal matrix.

c.) Prove that every positive definite matrix $K$ has a unique positive definite square root.

How will I be able to prove these?

  • For part a - I know I must use Gram matrix $K = A^TA$ in order to prove it.

  • For part c - I do not know what to do.

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How is this question b.) with the Jordan basis? What do you mean by that? – Berci Nov 26 '12 at 22:41
@Berci I dont quite understand your question. Do you mean my attempt at a solution or the question? – diimension Nov 26 '12 at 22:47
up vote 2 down vote accepted

The classic way for doing a) is to use the singular value decomposition. Then $A=UDV$, with $D$ diagonal with non-negative diagonal entries, and $U,V$ orthogonal. Since $A$ is invertible, $D$ has to be invertible, so it is positive-definite. Now let $Q=UV$, $B=V^*DV$. Then $B$ is positive-definite and $Q$ is orthogonal.

If $\det A>0$, then $\det Q>0$ (since $\det B>0$). So $\det Q=1$.

I have to admit that I cannot understand what is being asked in question b).

As for c), is $K$ is positive-definite then (under the usual assumption that $K$ is symmetric), $K=VDV^*$, with $V$ orthogonal and $D$ diagonal and positive definite. Let $E$ be the diagonal matrix with $E_{kk}=\sqrt{D_{kk}}$. Then $E$ is positive-definite and $E^2=D$. Now put $L=VEV^*$; then $L$ is positive-definite and $L^2=K$.

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Thank you very much, Martin! I will edit out part b. because you and Berci told me the same thing. – diimension Nov 27 '12 at 0:11

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