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spasmous
science forum addict
Joined: 03 May 2005
Posts: 66
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 Posted: Fri Jun 23, 2006 11:18 pm Post subject:
Is A'A always postive semidefinite?
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As per title, is A transpose times A always postive semidefinite for
all matrices? Is there a standard reference that I can refer to? Thanks. |
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Roy Stogner
science forum beginner
Joined: 13 Jun 2005
Posts: 38
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| Posted: Fri Jun 23, 2006 11:37 pm Post subject:
Re: Is A'A always postive semidefinite?
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On Fri, 23 Jun 2006 16:18:34 -0700, spasmous wrote:
| Quote: | As per title, is A transpose times A always postive semidefinite for
all matrices? Is there a standard reference that I can refer to? Thanks.
|
Yes, and it's easy to derive. For any eigenpair g,x of A'A:
A'Ax = g*x
x'A'Ax = x'*g*x
(Ax)'*(Ax) = g*x'*x
nonnegative number = g * nonnegative number
---
Roy Stogner |
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Ray Koopman
science forum Guru Wannabe
Joined: 25 Mar 2005
Posts: 216
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| Posted: Fri Jun 23, 2006 11:56 pm Post subject:
Re: Is A'A always postive semidefinite?
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For any vector x, let Ax = v. Then x'A'Ax = v'v >= 0.
spasmous wrote:
| Quote: | As per title, is A transpose times A always postive semidefinite for
all matrices? Is there a standard reference that I can refer to? Thanks. |
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Sun Jun 25, 2006 8:35 am Post subject:
Re: Is A'A always postive semidefinite?
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In article <1151104714.550508.120250@b68g2000cwa.googlegroups.com>,
spasmous <spasmous@gmail.com> wrote:
| Quote: | As per title, is A transpose times A always postive semidefinite for
all matrices? Is there a standard reference that I can refer to? Thanks.
|
For all matrices with real entries, yes. For matrices with complex
entries, you need to use the Hermitian conjugate.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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