[R] General expression of a unitary matrix

[J. Liu]

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 at 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...
> > 
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> -- 
> Spencer Graves, PhD
> Senior Development Engineer
> PDF Solutions, Inc.
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> 
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