Let A be a Hermitian matrix with eigenvalues d1 > \2 > · .· > dn and orthonormal eigenvectors u1,..., Un. For any nonzero vector x E C", we define p(x) = (Ax, x) = x" Ax. (a) Let x = c¡u] + .cnun. Show that p(x) = |c1|²A1 + |c2|² \2 + • ..+ |cn[²An- (In particular, this formula implies p(u;) = A; for1
Let A be a Hermitian matrix with eigenvalues d1 > \2 > · .· > dn and orthonormal eigenvectors u1,..., Un. For any nonzero vector x E C", we define p(x) = (Ax, x) = x" Ax. (a) Let x = c¡u] + .cnun. Show that p(x) = |c1|²A1 + |c2|² \2 + • ..+ |cn[²An- (In particular, this formula implies p(u;) = A; for1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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q7
![7. Let A be a Hermitian matrix with eigenvalues A1 > A2 > ·· > \n and orthonormal eigenvectors
u1, ..., un. For any nonzero vector x E C", we define
p(x) 3 (Ах, х)
: хН Ах.
(a) Let x = cqu1+.. CnUn. Show that
p(x) = |c1²A1 + |c2]²\2 + · .. +
(In particular, this formula implies p(u;) = X; for 1<i<n.)
(b) Show that if x is a unit vector, then
An S p(x) < A1.
(This implies that if we view p(x) as a function defined on the set {x € C" | |x| = 1} of
unit vectors in C", it achieves its maximum value at uj and minimum value at un.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa54c3a1b-c0c6-4618-b897-4acbc63b0ce8%2Fd21b1e2c-43a8-4505-9967-045f6c65f0a9%2F3st7o2a_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let A be a Hermitian matrix with eigenvalues A1 > A2 > ·· > \n and orthonormal eigenvectors
u1, ..., un. For any nonzero vector x E C", we define
p(x) 3 (Ах, х)
: хН Ах.
(a) Let x = cqu1+.. CnUn. Show that
p(x) = |c1²A1 + |c2]²\2 + · .. +
(In particular, this formula implies p(u;) = X; for 1<i<n.)
(b) Show that if x is a unit vector, then
An S p(x) < A1.
(This implies that if we view p(x) as a function defined on the set {x € C" | |x| = 1} of
unit vectors in C", it achieves its maximum value at uj and minimum value at un.)
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