<http://search.barnesandnoble.com/OopBooks/UsedBook.asp?userid=55I6Q0XCIO&ean=2780923508552>
<http://search.barnesandnoble.com/OopBooks/UsedBook.asp?userid=55I6Q0XCIO&ean=2809402761018>
On Tue, 10 Jun 2003 17:58:25 -0400, mike law wrote:
>I have a very nice book, still in print, - i think
>
>ELEMENTARY LINEAR ALGEBRA WITH DERIVE, An Integrated Text by J. Hill
>& T. Keagy ISBN 0862384036 Chartwell LTD Chapter 5 is about 40 pages
> Eigenvalues and Eigenvectors, The Cayley Hamilton Theorem, Special
>Types of Matrices,Bounds for Eigenvalues,Jordan Canonical Form
>
>This book is 389 pages and begins elementary and is full of examples
>all for a reasonable price ..
>
>************************************************** On flâne sur
>l'avenue de Jean-Jaurès, et comme Paul Langevin, on essaie de se
>poser des questions.
>--- Mike Law ----- Original Message -----
>From: "Valeriu Anisiu" <[log in to unmask]> To: <DERIVE-
>[log in to unmask]> Sent: Tuesday, June 10, 2003 9:15 AM Subject:
>Re: eigenvectors
>
>
>Hello Marcelo,
>
>the problem of finding numerically the eigenvalues and specially the
>eigenvectors of a matrix is very difficult.
>The main reason is that even if the eigenvalues depend continuously
>on the matrix (as complex numbers), the eigenvectors (and their
>number) do not!
>Take for example the matrix M(a,b):=[1,a,0;0,1,b;0,0,1].
>M(a,b) has only one eigenvalue (1) but for any nonzero number eps:
>M(eps,0) has 2 eigenvectors: [1,0,0], [0,0,1]
>M(0,eps) has 2 eigenvectors: [1,0,0], [0,1,0]
>M(0,0) has 3 eigenvectors: [1,0,0], [0,1,0],[0,0,1].
>If we know only an approximation of a matrix A, it is impossible to
>give approximations for its eigenvectors!
>However, if A has distinct eigenvalues then the situation changes:
>the eigenvectors can be approximated but using several precautions
>because the linear system (A-kI)x=0 is very unstable. If k is an
>eigenvalue then det(A-kI)=0 but because of the roundoff errors, the
>determinant could be evaluated as nonzero and so x=0!
>
>In Derive, the following method is used by APPROX_EIGENVECTOR to
>compute the eigenvector corresponding to the eigenvalue k: x := SIGN
>( (A-kI)^(-1).b ) where b is a kind of arbitrary (random) vector.
>(in fact ROW_REDUCE is used to invert A-kI).
>It is assumed that k is only an approximation of the eigenvalue,
>otherwise (A-kI) would be singular.
>This approach is justified by the fact (from operator calculus) that
>if b is not in the range of A-kI, then LIM( SIGN( (A-tI)^(-1).b ),
>t, k) exists and it is an eigenvector.
>A problem appears when k is too close to the exact eigenvalue.
>You will have to apply a small perturbation or increase
>PrecisionDigits.
>For example, use APPROX_EIGENVECTOR(A, APPROX(k,PrecisionDigits-2)).
>Another (less likely) problem appears if b is in the range of A-kI:
>you will have to change it.
>Of course there exists better methods to compute eigenvectors but
>APPROX_EIGENVECTOR has the advantage of being very simple.
>
>P.S. The programs in eigenvals.dfw have several bugs and I do not
>recommand to use them; for example EIGENVALS(M(0,0.01)) approximates
>to [1,100,1].
>
>I hope that this rather long answer will help.
>
>Cheers, Valeriu
>
>
>
>
>
>
>_____________________________________________________________________
>_ Do you want a free e-mail for life ? Get it at
>http://www.personal.ro/
--
Patrick West, [log in to unmask] on 06/10/2003
<[log in to unmask]> <www.fpwest.com>
|