Eigenvectors

[Luke Setzer]

I am learning about eigenvalues and eigenvectors on the TI-Nspire. My
textbook walks through this lengthy process for generically solving
the eigenvectors for a linearly dependent matrix like this one:

a:=[1,2,3;2,4,6;3,6,9]

I am getting correct answers for the characteristic polynomial (the
charpoly() function) and the eigenvalues (the eigvl() function) but
can make no progress at all with the eigenvectors (the eigvc()
function).

Does this last function only work with linearly independent matrices?

[Wayne]

Luke,

The eigenvector function works fine. In the case you are working
with, the eigenvalue 0 has a two-dimensional eigenspace. The first
and third columns of the returned eigenvector matrix are members of
that two-dimensional space. The second column is the (one dimensional)
eigenvector corresponding to the eigenvalue 14. Refer to any
standard text on elementary linear algebra to study these phenomena.
I would suggest that you use the eigenvalue and eigenvector functions
in the library linalgcas that is included with OS 1.7. These
functions give the exact eigenvalues and eigenvectors where possible.
The functions included in the base CAS give numerical approximations
in some cases. Using linalgcas makes the study of linear algebra much
easier, particularly in the case of repeated eigenvalues, as in your
example.

Wayne

[ArtBelmonte]

Hi Luke,

The eigvl command on the TI-Nspire or TI-Nspire CAS will return
decimal approximations to the eigenvalues of a matrix. In your case it
returns the list {1.00E-14, 14.00, 3.88E-15}. The actual eigenvalues
are 0, 14, 0. The eigenvalue 0 has algebraic multiplicity 2 (it is a
double root of the characteristic equation) and geometric multiplicity
2 (there are two linearly independent eigenvectors associated with
it). Similary the eigenvalue 14 has algebraic multiplicity 1 and
geometric multiplicity 1. The eigvc command returns decimal
approximations to unit eigenvectors that correspond to the eigenvalues
returned by eigvc. (The adjective "unit" means that the Euclidean
length of the vector is 1.)

If you have a TI-Nspire CAS, then its symbolic capabilities enable you
to work with exact eigenvalues and eigenvectors (for sufficiently nice
matrices, such as those typically given in examples in your textbook).
Wayne <wayn...@bellsouth.net> mentioned the linalgcas library from
Texas Instruments in a reply to your post. Indeed, its eigenvals
command will return a list of eigenvalues, in your case {0, 14}. But
it doesn't give corresponding algebraic multiplicities (2 and 1 in
this case).

As I mentioned in a post to this group last July (2009), I wrote a
free differential equations package for the TI-Nsipre CAS along with
documentation, videos, and keyboard & inline library shortcuts. The
package is called TAMUDFEQ 1.2. Here is the relevant web page.

http://calclab.math.tamu.edu/~belmonte/TAMUDFEQ/TAMUDFEQ.html

The evmt command (eigenvalue multiplicity table) in this package
returns the exact eigenvalues for your matrix with their corresponding
algebraic multiplicities. Then two invocations of the eigenvect
command from the linalgcas library return a full set of linearly
independent eigenvectors associated with these eigenvalues.

Download the package and watch the videos for starters. I might add
that I am teaching differential equations this term. Accordingly, here
is my class web page with lots more information. The favorite videos
of students are the so-called "exam videos" in which I show how to
chain together built-in and package commands to solve typical problems
from differential equations.

http://calclab.math.tamu.edu/~belmonte/2010a_m308.html

Ideally, if the coffee holds out, I'll make additional videos this
term that are more streamlined! I hope this information has helped.

Regards, - Art Belmonte

[Luke Setzer]

Thanks, guys. I have never had a formal class in linear algebra and
am starting a linear algebra class audit next week online. I am
trying to get a head start on the material. The linalgcas library
file will be a big help. I uploaded two files to the Files section
here for a matrix different from the initial one I posted:

EigenvectorMathXL.pdf
Eigenvectors.tns

I am still puzzling over the results from the linalgcas eigenvectors
function because it gave a 3x2 matrix answer to two of the three
eigenvalues where I only expected a 3x1 matrix.

Anyway, hopefully the teacher will be able to explain this to me as I
progress through the class.

[Nelson Sousa]

an eigenvalue can have any number of linearly independent
eigenvectors from 1 to the algebraic multiplicity of the eigenvalue as
a solution of the characteristic equation. The number of independent
eigenvalues is called the geometric multiplicity of the matrix.

One particularly important property is that if a n*n matrix has n
linearly independent eigenvectors it can be diagonalized; if it has
m<n eigenvectors it can only be transformed in a matrix which is
diagonal by blocks, being the number of blocks equal to the number of
eigenvectors.

But hey, I don't want to spoil your linear algebra course. you'll have
plenty of time to master all this stuff!

Cheers,
Nelson

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