# Re: [R] General expression of a unitary matrix

From: Spencer Graves <spencer.graves_at_pdf.com>
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


>>"http://mathworld.wolfram.com/SpecialUnitaryMatrix.html", where I
>>learned that a "special unitary matrix" U has det(U) = 1 in addition
>>to
>>the "unitary matrix" requirement that
>>
>> U %*% t(Conj(U)) == diag(dim(U)[1]).
>>
>> Thus, if U is a k x k unitary matrix with det(U) = exp(th*1i),
>>exp(-th*1i/k)*U is a special unitary matrix. Moreover, the special
>>unitary matrices are a group under multiplication.
>>
>> Another Google query led me to
>>"http://mathworld.wolfram.com/SpecialUnitaryGroup.html", which gives
>>a
>>general expression for a special unitary matrix, which seems to
>>require
>>three real numbers, not four; with a fourth, you could get a general
>>
>>unitary matrix.
>>
>> spencer graves
>>
>>J. Liu wrote:
>>
>>
>>>Hi, all,
>>>
>>>Does anybody got the most general expression of a unitary matrix?
>>>I found one in the book, four entries of the matrix are:
>>>
>>>(cos\theta) exp(j\alpha); -(sin\theta)exp(j(\alpha-\Omega));
>>>(sin\theta)exp(j(\beta+\Omega)); (cos\theta) exp(j\beta);
>>>
>>>where "j" is for complex.
>>>However, since for any two unitary matrices, their product should
>>
>>also
>>
>>>be a unitary matrix. When I try to use the above expression to
>>>calculate the product, I can not derive the product into the same
>>
>>form.
>>
>>>Therefore, I suspect that this may not be the most general
>>
>>expression.
>>
>>>Could you help me out of this? Thanks...
>>>
>>>______________________________________________
>>>R-help@stat.math.ethz.ch mailing list
>>>https://stat.ethz.ch/mailman/listinfo/r-help
>>
>>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
>>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

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Received on Mon Aug 15 02:12:47 2005

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