January 28th, 2017, 10:35 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,190 Thanks: 90  Positive Definite
Given a quadratic form: $\displaystyle Q=x'Ax=x_{i}A_{ij}x_{j}$, summation convention, A symmetric. Let P diagonalize A: $\displaystyle D=P^{1}AP$. Let $\displaystyle x=Py$. Then $\displaystyle Q=y'Dy=\lambda _{i}y_{i}^{2}$, which is positive for all y if $\displaystyle \lambda$ are all positive. Question: Why is $\displaystyle x'Ax$ positive for all x? 
January 30th, 2017, 03:35 PM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,190 Thanks: 90 
Range of values of x'Ax doesn't change under nonsingular linear substitution.

January 30th, 2017, 11:31 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 216 Thanks: 116 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
It isn't necessarily. However, when it is, this is the definition of a positive definite matrix. If you are assuming that $A$ is positive definite then this follows immediately from what you have written since all eigenvalues are nonnegative. If you aren't assuming this, then here is a simple counterexample which shows not every symmetric matrix is positive definite. $$\left\( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$$  

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