(c) Show that max p(x) —D А X#0 min p(x) = and %3D X#0

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 11AEXP
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Need Part C please.
28. Let A be a Hermitian matrix with eigenvalues 1 >
12 2
u1,..., un. For any nonzero vector x in R", the
Rayleigh quotient p(x) is defined by
> An and orthonormal eigenvectors
x"Ax
x#x
(Ах, х)
p(x) =
%3D
%3D
(x, x)
(a) If x = cju + + C,un, show that
..
p(x) =
(b) Show that
An S p(x) <1
(c) Show that
max p(x) = A1
X40
min p(x) = n
X#0
and
Transcribed Image Text:Need Part C please. 28. Let A be a Hermitian matrix with eigenvalues 1 > 12 2 u1,..., un. For any nonzero vector x in R", the Rayleigh quotient p(x) is defined by > An and orthonormal eigenvectors x"Ax x#x (Ах, х) p(x) = %3D %3D (x, x) (a) If x = cju + + C,un, show that .. p(x) = (b) Show that An S p(x) <1 (c) Show that max p(x) = A1 X40 min p(x) = n X#0 and
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