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...
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... > > ______________________________________________ > R-help at stat.math.ethz.ch 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 spencer.graves at pdf.com www.pdf.com <http://www.pdf.com> Tel: 408-938-4420 Fax: 408-280-7915
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... > > > > ______________________________________________ > > R-help at stat.math.ethz.ch 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 > > spencer.graves at pdf.com > www.pdf.com <http://www.pdf.com> > Tel: 408-938-4420 > Fax: 408-280-7915 > > ______________________________________________ > R-help at stat.math.ethz.ch mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! > http://www.R-project.org/posting-guide.html
Could you provide an example that can NOT be expressed in that form? spencer graves J. Liu wrote:> 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... >>> >>>______________________________________________ >>>R-help at stat.math.ethz.ch 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 >> >>spencer.graves at pdf.com >>www.pdf.com <http://www.pdf.com> >>Tel: 408-938-4420 >>Fax: 408-280-7915 >> >>______________________________________________ >>R-help at stat.math.ethz.ch 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 spencer.graves at pdf.com www.pdf.com <http://www.pdf.com> Tel: 408-938-4420 Fax: 408-280-7915