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Re: Characteristic Polynomials and Eigenvalues
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19381] Re: [mg19364] Characteristic Polynomials and Eigenvalues
*From*: "Mark E. Harder" <harderm at ucs.orst.edu>
*Date*: Mon, 23 Aug 1999 13:57:04 -0400
*Sender*: owner-wri-mathgroup at wolfram.com
Manuel,
If a matrix, A, has eigenvectors, x, and eigenvalues,L (scalars),
corresponding to those eigenvectors, then, by definition, A x= Lx. So, then
A x -Lx= (A -L I) x=0, where I = the Identity matrix. The theory of linear
equations, part of Linear Algebra, then requires that Det[ A -L I ]=0 (Det
is a Mathematica function for the determinant of a matrix), and this is
where the characteristic polynomial comes in: the Determinant on the lhs of
this equation is the characteristic polynomial, and so the equation says
that the roots of this polynomial are the eigenvalues for A. (Use
Mathematica to show this for your matrix) It really has nothing to do with
straight lines or curves; this is algebra, not geometry (although there are
geometric applications).
To understand the theory of linear equations and the finer points of
eigensystem theory, you really do have to understand some linear algebra,
and I think you are going to have to read some book on the subject. There
is more than one book that takes you through linear algebra with the help of
Mathematica. See the books section of the Wolfram website. Also, other
books not involving Mathematica per se are "Applied Linear Algebra", by
Noble & Daniel, which is geared to applications, of course. And there is
always the Scaum's Outline for linear algebra, with lots of worked & solved
problems. Hope this helps.
-mark
-----Original Message-----
From: MAvalosJr at aol.com <MAvalosJr at aol.com>
To: mathgroup at smc.vnet.net
Subject: [mg19381] [mg19364] Characteristic Polynomials and Eigenvalues
>Gentlemen:
>
>I have been studying linear algebra and with the aid of several programs
and
>add- ons to Mathematica the task has been a piece of cake. However, the
time
>comes when suddenly "understanding" leers its ugly head.
>Given the vectors {4,-6}, {3, -7}, the characteristic polynomial is x^2 + 3
x
>-10. The eigenvalues are (-5,2), the eigenvectors are (2,3) and (3,1). My
>question:
>What does the characteristic polynomial (since it discribes a curve) have
to
>do with the vectors (which are straight lines)? Or for that matter, the
>eigenvalues and eigenvectors -derived from the matrix or the polynomial
have
>to do with the vectors?
>I plotted the polynomial but can't figure out what it has to do with the
>vectors.
>
>Thanks for whatever
>Manuel
>
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