Eigen values are not a good measure of Positive definiteness of a matrix

A matrix cannot be concluded as to whether it is positive definite or not using the eigen values alone.This can be proved using a counter example which is as follows.

This matrix has a has eigen values 1 and 2. but for x = 1, y =1 the value of the X'AX is -97

This counter example clearly proves that eigen values cannot be used to conclude whether a matrix is positive definite or not. Nevertheless it can be used when the matrix is symmetric.

PS: Discussion with T.R. Maruthi, MS scholar, EE Dept, was the impetus for this blog. Our discussion helped me know about this.