NetNews Usenet Archive 1992 #19/NN_1992_19.iso/spool/sci/math/10587

Text File | 1992-08-25 | 1.7 KB | 51 lines

Newsgroups: sci.math
Path: sparky!uunet!munnari.oz.au!manuel!rsphy7.anu.edu.au!rwc124
From: [e-mail] (Roderick Vance)
Subject: Matrix Logarithms
Message-ID: <[e-mail]>
Sender: [e-mail]
Reply-To: [e-mail] (Roderick Vance)
Organization: Optical Sciences Centre, Australian National University
Date: Wed, 26 Aug 92 05:05:44 GMT
Lines: 39

Let U be a unitary, symmetric matrix; U is expressible as

U = exp(i H), where H is real and symmetric.

I should like to know whether there is a simple method of finding
H given U, aside from diagonalising U. My aim is to find out, as simply as
i can, whether there are noughts in certain given positions of the H-matrix.

My present method is to diagonalise U = P L P^-1, where L is diagonal, and
then i get H = -i P log(L) P^-1. This method is forbiddingly unwieldy for
matrices larger than 4x4. I have thought of writing U as a power series in
H (the power series is finite, owing to the Cayley-Hamilton Theorem) but
cannot seem to make further headway from there.

A related problem is the following:

Let U(z) = exp(i H z) where H is real and symmetric and KNOWN and z is real.
Let x0 be a constant complex element vector. Then consider the set:

T = {a U(z) x0: a is complex, z is real}

This time, GIVEN H, is there a simple test to find out whether another
given vector
xf belongs to T.

Note: It is dead simple to show that T is NOT a linear space, so the
problem isn't
as easy as finding a basis spanning T.

Anyone interested in this problem: please post answers on the net or
write to me
directly, as you will.

Thanks in anticipation of your answers.


Roderick Vance
Optical Sciences Centre
Australian National University