# Math Help - quadratic form and its eigenvalues

1. ## quadratic form and its eigenvalues

The question is: Consider the quadratic form $x^TAx$ where A is a k x k matrix and x is a k x 1 vector. Show that if A is positive definite, then all its eigenvalues are positive.

Not sure where to start.

2. Let $\lambda$ be an eigenvalue of A and let x be an non-zero eigenvector corresponding to $\lambda$. What is $x^TAx$?

3. Originally Posted by garymarkhov
The question is: Consider the quadratic form $x^TAx$ where A is a k x k matrix and x is a k x 1 vector. Show that if A is positive definite, then all its eigenvalues are positive.

Not sure where to start.

Let $\lambda\,\,and\,\,v$ be an eigenvalue of $A$ and one of its corresponding eigenvectors, then:

$00\,\,\,as\,\,\,x^tx>0\,\,\,\forall\,\,\,ve ctor\,\,\,x \neq 0$

Tonio