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.

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- Dec 11th 2009, 05:27 AMAvi06Symmetric Matrix proof !
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. - Dec 11th 2009, 08:11 AMHallsofIvy
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, .