Given that A is a n x n symmetric matrix and , where are eigenvalues of A.
Show that:
Having trouble proving the above ! Any help would be much appreciated.
Since A is symmetric, it is diagonalizable. That is, there exist orthogonal P such that where D is the diagonal matrix with the eigenvalues of A on the diagonal. In that case, so that or, letting y= Px, . Now it is clear that which is necessarily larger than and smaller than .
Finally, since P is orthogonal, .