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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
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