| Science Forum Index » Mathematics Forum » A*x <= h*x ? x is random vector, A is square matrix, h... |
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| hhwolf76... |
 Posted: Thu Nov 05, 2009 10:58 am |
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I got a question. If A is a square matrix, h is one of
A's eigenvalues. If x is the corresponding eigenvector,
then A*x=h*x. But if x is not eigenvector, it is a random vector, is it true that A*x<=h*x ?
Not familar with the property of eigenvector. Thanks! |
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| hhwolf76... |
| Posted: Thu Nov 05, 2009 11:02 am |
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Or in other way,
Is it true that x'*A*x<=x'*h*x ?
Still, A is square matrix, h is eigenvalue, x is random vector. |
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| hhwolf76... |
| Posted: Thu Nov 05, 2009 4:23 pm |
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I assume A=R'*R (R is a square matrix),
h is the biggest eigenvalue.
Is my assumption right? |
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| Ken Pledger... |
| Posted: Thu Nov 05, 2009 6:04 pm |
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In article
<854775835.17564.1257454992142.JavaMail.root at (no spam) gallium.mathforum.org>,
hhwolf76 <james.zhou76 at (no spam) gmail.com> wrote:
[quote]Or in other way,
Is it true that x'*A*x<=x'*h*x ?
Still, A is square matrix, h is eigenvalue, x is random vector.
[/quote]
Try A =
(1 0)
(0 2)
with h = 1 and x = (0 1)
Ken Pledger. |
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| Ray Vickson... |
| Posted: Sat Nov 07, 2009 7:23 am |
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On Nov 5, 6:23 pm, hhwolf76 <james.zho... at (no spam) gmail.com> wrote:
[quote]I assume A=R'*R (R is a square matrix),
h is the biggest eigenvalue.
Is my assumption right?
[/quote]
Yes. All you need is for A to be real and symmetric. See
http://en.wikipedia.org/wiki/Rayleigh_quotient and
http://www.umiacs.umd.edu/~shaohua/enee739q_cmsc858c/RayleighsQuotient.pdf
..
This last link shows that a stationary point of the Rayleigh quotient
yields an eigenvalue, and since the max/min of the rayleigh quotient
corresponds to a stationary point, these max and min ratios are the
largest and smallest eigenvalues.
R.G. Vickson |
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