Re: [R] General expression of a unitary matrix

From: J. Liu <liuj24_at_univmail.cis.mcmaster.ca>
Date: Mon 15 Aug 2005 - 00:36:16 EST

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:
> "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...
> >
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> --
> Spencer Graves, PhD
> Senior Development Engineer
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