Hello all Derivers,

Rhombus still has some copies in stock of this book.

See http://www.rhombus.be/Books/derive.htm

Best regards,

Rosette Van den Wouwer
Zaakvoerder Rhombus b.v.b.a.

Tel : 03 605 7891
Fax : 03 605 7892

www.rhombus.be


-----Original Message-----
From: DERIVE computer algebra system [mailto:[log in to unmask]]On
Behalf Of Philip Yorke
Sent: Wednesday, June 11, 2003 11:45 AM
To: [log in to unmask]
Subject: Re: eigenvectors

Following Patrick's recommendation for the book "Elementary Linear
Algebra with Derive" by Bob Hill and Tom Keagy, if any of you are
trying to trace it you are unlikely to obtain it through conventional
channels.  The last I heard was that the whole stock of the book is in
the hands of the authors, probably reachable through Duquesne
University, USA. (Sadly I don't have current email addresses for them.)
Cheers
Philip Yorke

Chartwell-Yorke Ltd.
114 High Street, Belmont Village, Bolton, Lancashire, BL7 8AL
England, tel (+44) (0)1204 811001, fax (+44) (0)1204 811008
[log in to unmask], www.chartwellyorke.com

Chartwell-Yorke Ltd. is a member of EMSET

On Tuesday, June 10, 2003, at 11:53  pm, Patrick West wrote:

> <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