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 Post subject: Proof: Eigenvalues of Matrix A*APosted: Tue, 16 Dec 2008 20:20:35 UTC
 Math Cadet

Joined: Tue, 16 Dec 2008 20:15:02 UTC
Posts: 7
Hey all,

I have to prove that all eigenvalues of the matrix A*A are non-negative real numbers (where A* denotes the conjugate transpose matrix of A). How do you show this? Thank you for any help!

Baz

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 Post subject: Re: Proof: Eigenvalues of Matrix A*APosted: Tue, 16 Dec 2008 23:17:14 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 7 Jan 2006 18:29:24 UTC
Posts: 1401
Location: Leeds, UK
Bazzman wrote:
I have to prove that all eigenvalues of the matrix A*A are non-negative real numbers (where A* denotes the conjugate transpose matrix of A). How do you show this?

If A*Ax = λx then . So .

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 Post subject: Posted: Thu, 18 Dec 2008 16:18:31 UTC
 Math Cadet

Joined: Tue, 16 Dec 2008 20:15:02 UTC
Posts: 7
Thank Opalg.

Sorry I didnt reply earlier but I fell sick. Yes that makes sense - and they are real b/c of inner product <Ax,Ax> which would square all the complex terms in Ax.

I had two questions however if you dont mind. Im a bit of a noob with using the <,> notations and tend not to use it. So if you were to write <Ax,Ax> out how would it look (as in x*A*Ax,...)??

This would also probably answer my second question but why did you begin to work with <Ax,Ax>.

Thanks if you can answer.

Baz

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 Post subject: Posted: Thu, 18 Dec 2008 16:47:09 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 7 Jan 2006 18:29:24 UTC
Posts: 1401
Location: Leeds, UK
Bazzman wrote:
Im a bit of a noob with using the <,> notations and tend not to use it. So if you were to write <Ax,Ax> out how would it look (as in x*A*Ax,...)??

This would also probably answer my second question but why did you begin to work with <Ax,Ax>.

These are alternative notations for the same thing. Analysts usually think in terms of inner products and write . Algebraists think in terms of matrices and write . But these notations both mean exactly the same thing.

Choose whichever one you feel more at home with. As you can see, I'm an analyst.

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 Post subject: Posted: Thu, 18 Dec 2008 17:28:42 UTC
 Math Cadet

Joined: Tue, 16 Dec 2008 20:15:02 UTC
Posts: 7
Hi Opalg,

I realized they were alternative notations, yet I did not know who uses which type. My question however was how would you express your "analyst" equations in terms of "algebraic" equations?

<A*Ax,x> = x*A*Ax

(lambda)<x,x> = (lambda)x*x

How would <Ax,Ax> look/fit in?

Baz

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 Post subject: Re: Proof: Eigenvalues of Matrix A*APosted: Thu, 18 Dec 2008 20:15:10 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 7 Jan 2006 18:29:24 UTC
Posts: 1401
Location: Leeds, UK
Okay, if A*Ax = λx then . So .

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 Post subject: Posted: Thu, 18 Dec 2008 21:26:13 UTC
 Math Cadet

Joined: Tue, 16 Dec 2008 20:15:02 UTC
Posts: 7
Thank you very much for your time and help!

Baz

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