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

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

• Look at the associated quadratic form, and repeatedly "complete the square". – Lord Shark the Unknown May 10 '18 at 6:43
• Not every positive definite matrix is orthogonal. – Sean Roberson May 10 '18 at 6:44
• 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 6:56
• You may also repeatedly "remove squares" (as opposed to completing square). This results in Cholesky decomposition. – user1551 May 10 '18 at 7:01