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From: jfh@browngr.UUCP (John "Spike" Hughes)
Newsgroups: net.math
Subject: Re: Some Math problems
Message-ID: <1493@browngr.UUCP>
Date: Sun, 28-Oct-84 23:48:16 EST
Article-I.D.: browngr.1493
Posted: Sun Oct 28 23:48:16 1984
Date-Received: Tue, 30-Oct-84 19:30:23 EST
References: dartvax.2503, <689@ihuxt.UUCP>
Lines: 25

There's a general method for finding the square root of
a symmetric matrix: Every real symmetric matrix has
real non-negative eigenvalues (see the definition of eigenvalue in any
linear algebra book). Let the eigenvalues be a1, a2, ..., an,
and corresponding unit eigenvectors be v1, v2, ..., vn (this means
that:
       M vn = an vn

(here M is the matrix in question). Now line up all the eigenvectors
in a matrix, each eigenvector being a column, and call the matrix Q. Then
Qt M Q is a diagonal matrix D (with the eigenvalues of M down the
diagonal) (here Qt denotes the transpose of Q). Let E be the square
root of D (i.e. take the square root of each entry in D). Then a
square root of M is given by

     Q E Qt

This method also works for non-symmetric matrices, if they have
all positive eigenvalues, and all are real, but you must use
Q inverse rather than Q transpose (alas).
     There is also a polar coordinate decomposition of a matrix
into the product of a real symmetric matrix R and a unitary matrix
U (in the case of a 2*2 complex matrix, these correspond to
r and theta for polar coordinates in the plane), and you can do something
from there, but you probably don't wnat to...\