What is the proof behind positive definite matrix $A=QQ^T$?

Asked May 10 '18 at 06:40

Active May 10 '18 at 06:55

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I read this a lot of times, but I can't seem to prove it. How does a positive definite matrix $A$ decompose to $QQ^T$?

edited May 10 '18 at 06:55

asked May 10 '18 at 06:40

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1

Look at the associated quadratic form, and repeatedly "complete the square". – Angina Seng May 10 '18 at 06:43
Not every positive definite matrix is orthogonal. – Sean Roberson May 10 '18 at 06:44
3

Use the spectral decomposition and then take the square roots of the (positive) eigenvalues. Take a look at this. – Rodrigo de Azevedo May 10 '18 at 06:56
2

You may also repeatedly "remove squares" (as opposed to completing square). This results in Cholesky decomposition. – user1551 May 10 '18 at 07:01

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