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.

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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