[R] General expression of a unitary matrix

[Spencer Graves]

	  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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