Let A be a Hermitian matrix with eigenvalues A1 A2 > An and orthonormal eigenvectors U₁,...,Un. For any nonzero vector X ECn, we define p(x)= (Ax,x) = XHAX. Show that if x is a unit vector, then λη

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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Let A be a Hermitian matrix with eigenvalues A₁ A₂ >...>^n and
orthonormal eigenvectors U₁,..., Un. For any nonzero vector X ECn, we
define
p(x)= (Ax,x) = XHAX.
Show that if x is a unit vector, then
λη <p(x) ≤λι.
Transcribed Image Text:Let A be a Hermitian matrix with eigenvalues A₁ A₂ >...>^n and orthonormal eigenvectors U₁,..., Un. For any nonzero vector X ECn, we define p(x)= (Ax,x) = XHAX. Show that if x is a unit vector, then λη <p(x) ≤λι.
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