Re: Solving a matrix equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg44350] Re: Solving a matrix equation*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 5 Nov 2003 10:02:19 -0500 (EST)*Organization*: The University of Western Australia*References*: <bo7o1n$aeh$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <bo7o1n$aeh$1 at smc.vnet.net>, josegomez at gmx.net wrote: > Let T and Q be two nxn complex matrices (T is Hermitian, Q is > not). I want to test whether a vector p (nx1) is an eigenvector of the > following combination: > A=(Inverse[T]*Q)*(Inverse[T]*Conjugate[Transpose[Q]]), > or, in LaTeX form: > (T^{-1}Q)*(T^{-1}*Q^{*T}. I assume you mean (dropping the unnecessary parentheses) A = Inverse[T].Q.Inverse[T].Conjugate[Transpose[Q]] ? Note that Conjugate does not work well with symbolic parameters. Instead, the replacement rule m /. Complex[x_, y_] :> Complex[x, -y] is a simple way to obtain the (symbolic) conjugate of an object m, appropriate in most situations. Also, I define SuperDagger[m_?MatrixQ]:=Transpose[m]/.Complex[x_, y_]:>Complex[x, -y] (which formats as, and can be input as, a superscripted dagger) for such applications. > I have punched the previous lines into Mathematica, and tried > to see whether my vector p was an eigenvector. Testing whether something is an eigenvector should be completely straightforward. Just substitute it in. > However, I had Mathematica eat up all the memory and subsequently crash, so I am > asking here to see whether someone can suggest a way around this. > > The problem is 3x3 (n=3), but due to the relatively large number > of parameters, it is complicated and error-prone to do by hand. When computing eigenvalues and eigenvectors of matrices it is useful to use SetOptions[Roots, Cubics -> False, Quartics -> False] to prevent Mathematica from explicitly solving the Cubics that arise when computing the roots of a cubic equation (for n=3). This should greatly reduce the memory usage. Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul