Thread: a real invertible matrix such that PtAP is diagonal

1. a real invertible matrix such that PtAP is diagonal

I was reading this book and it says

determine a real invertible matrix P such that P^TAP is diagonal

where

matrix A is

$\displaystyle [(0 , 1 , 2), (1 , 0 , 0 ),(2 , 0 , 0)]$

I want to know how to find P.
The text book skipped the steps that i want.
I want to know the steps.
Thank you

2. Originally Posted by Livevvire
I was reading this book and it says

determine a real invertible matrix P such that P^TAP is diagonal

where

matrix A is

$\displaystyle [(0 , 1 , 2), (1 , 0 , 0 ),(2 , 0 , 0)]$

I want to know how to find P.
The text book skipped the steps that i want.
I want to know the steps.
Thank you
Find the eigenvectors of $\displaystyle A$ and set these eigenvectors as the columns of matrix $\displaystyle P$.

Edit: I re-read your original post and it's asking for a different method than I'm used to.

If you use the method I've given, then $\displaystyle P^{-1}AP$ is diagonal, not $\displaystyle P^TAP$...

3. I posted a similar question (see thread, Diagonalizing Symmetric Bilinear Forms a few threads below).

Thing is though, my notes tells me i have to scale these by the length of the eigenvector, but Prove It hasn't mentioned that in his post. Should 'P' be scaled or can it be left as the columns being the eigenvectors?