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

From: Spencer Graves <spencer.graves_at_pdf.com>
Date: Sun 14 Aug 2005 - 02:06:23 EST

"
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)).

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
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