Let A be a symmetric n n matrix with eigenvalues 1, . . . , n and orthonormal eigenvectors u1, . . . , un. Let x Rn and let ci = uT i x for i = 1, 2, . . . , n. Show that (a) _Ax_22 = _n i=1 (i ci )2 (b) If x _= 0, then min 1in |i| _Ax_2 _x_2 max 1in |i (c) _A_2 = max 1in |i | 2

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