The idea of using row reduction for the eigenvalue problem is new to
me, and I like the idea, and your paper, a lot. Thank you.
I am having some trouble trying the code.
1) I tried to load the demo file, but it failed with the message
"Could not find `c:/maxima/matrixoperations/matrixops.mac' ..."
I am running Linux.
2) I tried to load linalg.mac on my Knoppix CD (that is what I use
in my classes), but it could not find the nset package - probably
because Maxima on Knoppix CD is 5.9.0.
3) linalg.mac loaded in both Maxima 5.9.1 and Maxima 5.9.0rc2 (two
different computers), but I got a full screen of warning on Maxima
5.9.1. I tried your example below, but ptriangularize(m,z) failed
with the message
Argument to EMPTYP must be a list.
#0:
ptriangularize(m=MATRIX([1-z,2,3],[4,5-z,6],[7,8,9-z]),v=z)(linalg.mac
line 196)
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);
If you think that it might be of any help to you, I could try this
again, and copy the warning/error messages in the e-mail ...
Milan Lukic
Barton Willis [03/05/05 14:56 -0500]:
> I wrote code for basic linear algebra computations --- nullspace,
> column space, and orthogonal complements. You can get it
> from
>
> http://www.unk.edu/acad/math/people/willisb/home.html
>
> In the file linearalgebra.zip, you'll find demo, usage, and test files.
>
> Additionally, there is code for finding the spectrum that
> only uses row reduction. For a description, see
>
> http://www.unk.edu/acad/math/people/willisb/eigens-by-row.pdf
>
> An example
>
> (%i1) load("l:/linearalgebra/linalg.mac")$
> Warning - you are redefining the Macsyma function rank
>
> (%i2) m : matrix([1,2,3],[4,5,6],[7,8,9]) - z * ident(3);
> (%O2) matrix([1-Z,2,3],[4,5-Z,6],[7,8,9-Z])
> (%i3) algebraic : true$
> (%i4) ptriangularize(m,z);
> (%O4)
> matrix([4,5-Z,6],[0,66/49,-Z^2/7+(102*Z)/49+132/49],[0,0,(49*Z^3)/264-(245*Z^2)/88-(147*Z)/44])
>
> Tell maxima that the 3,3 entry is zero. Thus z is any eigenvalue.
>
> (%i5) tellrat(%[3,3]);
> (%O5) [Z^3-15*Z^2-18*Z]
> (%i6) ratsimp(%o4);
>
> The eigenspace corresponding the eigenvalue z is
>
> (%O6) matrix([4,5-Z,6],[0,66/49,-(7*Z^2-102*Z-132)/49],[0,0,0])
> (%i7) nullspace(%);
> (%O7) SPAN(matrix([1],[(Z^2-14*Z-16)/8],[-(Z^2-18*Z-12)/12]))
>
> where z is any eigenvalue.
>
> Barton
>
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