From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Matrices
Date: 6 Apr 1998 16:04:11 GMT

In article <352827C9.2865@xroadstx.com>,
Charlie Higgins  <chh@xroadstx.com> wrote:

>My problem is that the new math books that are out (or are being printed
>now) really stress matrices.  I know the simple rigors of matrices but
>not having had linear algebra in college thats about it.  I had a
>student ask me the other day "What is an eigenvalue?"  I had heard of it
>but could not explain to her what it was.  I told her I would find out. 

The emphasis on matrices per se is lamentable. It would be much better
if the books would focus on the underlying linear transformations. 

Geometrically, these are the functions on the plane, say, for which the
image of every parallelogram-at-the-origin is another 
parallelogram-at-the-origin, including rotations, scalings, shear maps,
and combinations of those. You can express these using a matrix
_once a coordinate system is introduced_, but the introduction of a
coordinate system is a little artificial here, and certainly hides
the underlying geometry.

This also helps explain eigenvalues. Suppose you wanted to stretch the
plane so that the unit circle at the origin was transformed into an
ellipse. This transformation tends to twist most vectors at the origin,
but the ones which point along the major and minor axes are simply
stretched to be multiples of themselves, not rotated at all. These
vectors are called eigenvectors, and the amounts by which they are
stretched -- say  a  and  b  -- are called eigenvalues. Note that
if two of _these_ vectors are used to establish coordinates, then the
linear transformation is simply  

           [ x ]     [ a  0  ] [ x ]
	L( [   ] ) = [       ] [   ] ,
           [ y ]     [ 0  b  ] [ y ]

that is, it is described by multiplication by a _diagonal_ matrix; a
similar expression for  L  is possible with respect to any other
coordinate system, but you wouldn't get anything so simple as a
diagonal matrix.

It is certainly valuable from a practical perspective to be comfortable
with a great many algebraic manipulations of matrices, but in my
experience it's difficult to know What Is Really Happening until you
express the results geometrically.

dave