From: Spencer Graves <spencer.graves_at_pdf.com>

Date: Mon 15 Aug 2005 - 02:05:19 EST

*>>"http://mathworld.wolfram.com/SpecialUnitaryMatrix.html", where I
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*>>learned that a "special unitary matrix" U has det(U) = 1 in addition
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*>>to
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*>>the "unitary matrix" requirement that
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*>>
*

*>> U %*% t(Conj(U)) == diag(dim(U)[1]).
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*>>
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*>> Thus, if U is a k x k unitary matrix with det(U) = exp(th*1i),
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*>>exp(-th*1i/k)*U is a special unitary matrix. Moreover, the special
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*>>unitary matrices are a group under multiplication.
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*>>
*

*>> Another Google query led me to
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*>>"http://mathworld.wolfram.com/SpecialUnitaryGroup.html", which gives
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*>>a
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*>>general expression for a special unitary matrix, which seems to
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*>>require
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*>>three real numbers, not four; with a fourth, you could get a general
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*>>
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*>>unitary matrix.
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*>>
*

*>> spencer graves
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*>>
*

*>>J. Liu wrote:
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*>>
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*>>
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*>>>Hi, all,
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*>>>
*

*>>>Does anybody got the most general expression of a unitary matrix?
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*>>>I found one in the book, four entries of the matrix are:
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*>>>
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*>>>(cos\theta) exp(j\alpha); -(sin\theta)exp(j(\alpha-\Omega));
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*>>>(sin\theta)exp(j(\beta+\Omega)); (cos\theta) exp(j\beta);
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*>>>
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*>>>where "j" is for complex.
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*>>>However, since for any two unitary matrices, their product should
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*>>
*

*>>also
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*>>
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*>>>be a unitary matrix. When I try to use the above expression to
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*>>>calculate the product, I can not derive the product into the same
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*>>
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*>>form.
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*>>
*

*>>>Therefore, I suspect that this may not be the most general
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*>>
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*>>expression.
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*>>
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*>>>Could you help me out of this? Thanks...
*

*>>>
*

*>>>______________________________________________
*

*>>>R-help@stat.math.ethz.ch mailing list
*

*>>>https://stat.ethz.ch/mailman/listinfo/r-help
*

*>>>PLEASE do read the posting guide!
*

*>>
*

*>>http://www.R-project.org/posting-guide.html
*

*>>
*

*>>--
*

*>>Spencer Graves, PhD
*

*>>Senior Development Engineer
*

*>>PDF Solutions, Inc.
*

*>>333 West San Carlos Street Suite 700
*

*>>San Jose, CA 95110, USA
*

*>>
*

*>>spencer.graves@pdf.com
*

*>>www.pdf.com <http://www.pdf.com>
*

*>>Tel: 408-938-4420
*

*>>Fax: 408-280-7915
*

*>>
*

*>>______________________________________________
*

*>>R-help@stat.math.ethz.ch mailing list
*

*>>https://stat.ethz.ch/mailman/listinfo/r-help
*

*>>PLEASE do read the posting guide!
*

*>>http://www.R-project.org/posting-guide.html
*

>

>

Date: Mon 15 Aug 2005 - 02:05:19 EST

Could you provide an example that can NOT be expressed in that form?

spencer graves

J. Liu wrote:

> Thank you, Spencer. I read through the websites you suggested. What I > need is how to parameterize a 2\times 2 unitary matrix. Generally, > since for a complex 2\times 2 matrix, there are 8 free variables, and > for it to be unitary, there are four constraints (unit norm and > orthogonality), hence I think there are four free variables left for a > 2\times 2unitary matrix. The form I found can not decribe all the > unitary matrix, that is why I suspect that it is not the most general > one. The form in the second web you suggested is an interesting one, > however, since only 3 variables invovled, it may not be the most > general expression. > > Jing > > > On Sat, 13 Aug 2005 09:06:23 -0700 > Spencer Graves <spencer.graves@pdf.com> wrote: >

>> Google led me to

>

>

-- Spencer Graves, PhD Senior Development Engineer PDF Solutions, Inc. 333 West San Carlos Street Suite 700 San Jose, CA 95110, USA spencer.graves@pdf.com www.pdf.com <http://www.pdf.com> Tel: 408-938-4420 Fax: 408-280-7915 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Mon Aug 15 02:12:47 2005

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