# Need Help with positive definite matrices

## Homework Statement

If A is positive definite, show that ## A = C C^T ## where ## C ## has orthogonal columns.

## The Attempt at a Solution

So, I've got the first part figured out. Because ## A ## is symmetric, an orthogonal matrix ## P ## exists such that ## P^TA P = D = diag(\lambda_1,...,\lambda_n) ## where ## \lambda_i > 0 ## because A is positive definite. Next I've defined ## B = diag(\sqrt{\lambda_1},...,\sqrt{\lambda_n}) ## Then I wrote ## C = P^TB P ## then ## C C^T = (P^T B P) (P^T B P)^T = (P^T B P) P^T B^T P = P^T B B P = P^T D P = A ##

So I'm at the last step and I'm stuck on how to show that C has orthogonal columns. Any hints?

## The Attempt at a Solution

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Office_Shredder
Staff Emeritus
Gold Member
C has orthogonal columns if and only if CtC is a diagonal matrix (you should be able to check that this claim is true easily- the off diagonal entries of Ct C are the inner products of the columns of C).

Also PDPt = A, not PtDP.

Right, that's just sloppyness on my part typing that out. I haven't learned anything about inner products yet. Is there a way to look at this differently?

Office_Shredder
Staff Emeritus