Re: eigenvalues / eigenvectors

There's a good visualization of eigenvectors/values, with a lecture,
at: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/Tools/

If you don't understand the basics of this lecture (i.e. about
matrices, determinants and real/imaginary values) you need to go
farther back and study them. It shouldn't take much time to understand
the basics well-enough to understand this lecture.

It's pretty tough to just open a book on linear algebra and get an
understanding of eigenavalues. I studied all this formally at one time
and have forgotten most of it. When I studied linear algebra, we
called it "mystery math".

About 2/3 way through the semester, one guy called out to the prof,
what are we doing? The prof was a nice guy and responded, do you mean
today, or in the course? When we all grunted, "in the course", he took
the time to explain what was going on.

Linear algebra has to be approached on a systematic basis by learning
a series of definitions. The purpose of linear algebra is basically to
manipulate the numerical values in equations in an attempt to find a
solution to simultaneous equations.

The good news is that it is only a 3 month course. If you want to
understand eigenvectors/values, you really need to start at the
beginning of a book on linear algebra and get a feeling for what they
are doing. It shouldn't take any more than a few days intensive study
to glean the basics. Our 3 month course comprised 3 hours per week,
and over 3 months, that's no more than 36 hours lecture time.

Be warned, however, that many mathematicians have a bent for speaking
in a language foreign to most of us. There are basically two
approaches to math: one invloves the old fashioned math where things
tend to make physical sense, and then theres the new math where
reality has been subverted to serve the mathematics entity.

If you open a book on linear algebra, and it doesn't make sense, put
it down and find one that does. If the author is speaking in 'new
math', that is, one space is contained in another, or one space is a
subset of another, I'd chuck it. A book on linear algebra should
start out by explaining in plain English that it is about solving
simultaneous equations and get down to it. There's no need for
mathematical gobbeldy gook.

The whole basis of linear algebra is to remove the numerical values
form equations and plug them into a matrix. Then you can operate on
them using certain rules to reduce them. It's quite simple, actually.
One of the tricks, however, and basic function of linear algebra is to
determine whether a solution exists.

On the other side of the coin is vector calculus. Linear algebra
doesn't deal with vectors per se. A vector needs both magnitude and
direction in it's definiton. We took linear algebra and vector
calculus back to back. You don't need to study an entire course of
vector calculus, but it helps to understand how vecors rotate and/or
translate in space. You do calculations on vectors using their scalar
quantities, and those reduce to simultaneous equations.



On Tue, 07 Mar 2006 10:56:34 -0600, "jquest"
<james.quest@xxxxxxxxxxxx> wrote:

Does anyone know of a good resource that explains this well?
Thanks

James


.

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