# quadratic form and its eigenvalues

• Oct 23rd 2009, 02:19 AM
garymarkhov
The question is: Consider the quadratic form $\displaystyle x^TAx$ 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.
• Oct 23rd 2009, 05:23 AM
HallsofIvy
Let $\displaystyle \lambda$ be an eigenvalue of A and let x be an non-zero eigenvector corresponding to $\displaystyle \lambda$. What is $\displaystyle x^TAx$?
• Oct 23rd 2009, 08:58 AM
tonio
Quote:

Originally Posted by garymarkhov
The question is: Consider the quadratic form $\displaystyle x^TAx$ 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 $\displaystyle \lambda\,\,and\,\,v$ be an eigenvalue of $\displaystyle A$ and one of its corresponding eigenvectors, then:

$\displaystyle 0<v^tAv=v^t(\lambda v)=\lambda v^tv \Longrightarrow \lambda>0\,\,\,as\,\,\,x^tx>0\,\,\,\forall\,\,\,ve ctor\,\,\,x \neq 0$

Tonio