Question: Let A be a Hermitian matrix with eigenvalues Î»1 ¥ Î»2 ¥ ... ¥ Î»n and orthonormal eigenvectors u1,...,un. For any nonzero vector x in

Let A be a Hermitian matrix with eigenvalues Î»1 ‰¥ Î»2 ‰¥ ... ‰¥ Î»n and orthonormal eigenvectors u1,...,un. For any nonzero vector x in Cn the Rayleigh quotient p(x) is defined by

(a) If x = c1u1 + ... + cnun show that

(b) Show that
Î»n ‰¤ p(x) ‰¤ Î»1

(x) = (Ax, x) = X"AX

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