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From: worley@Navajo.ARPA
Newsgroups: net.math
Subject: Re: Some Math problems
Message-ID: <111@Navajo.ARPA>
Date: Wed, 31-Oct-84 18:12:17 EST
Article-I.D.: Navajo.111
Posted: Wed Oct 31 18:12:17 1984
Date-Received: Sat, 3-Nov-84 05:38:25 EST
References: dartvax.2503, <689@ihuxt.UUCP> <1493@browngr.UUCP>
Organization: Stanford University
Lines: 18

> There's a general method for finding the square root of
> a symmetric matrix: Every real symmetric matrix has
> real non-negative eigenvalues ...

(-2) is a (trivial) 1X1 real symmetric matrix which does not
have non-negative eigenvalues. You need the matrix to be
real symmetric positive definite (i.e. explicitly require that
the eigenvalues be positive) in order to guarantee that
    B = Q(sqrt(D))Q(t)
    A = BB
will have a non-complex B.

> 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).

You also need a full set of eigenvectors, which don't always exist
for a nonsymmetric matrix.