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 <[EMAIL PROTECTED]> 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... > > > > ______________________________________________ > > [email protected] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-help > > PLEASE do read the posting guide! > 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 > > [EMAIL PROTECTED] > www.pdf.com <http://www.pdf.com> > Tel: 408-938-4420 > Fax: 408-280-7915 > > ______________________________________________ > [email protected] mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! > http://www.R-project.org/posting-guide.html ______________________________________________ [email protected] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
