Re: matrices with "repeated eigenvectors"
- From: Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 17 Aug 2009 11:23:28 -0500
A N Niel <anniel@xxxxxxxxxxxxxxxxxxxxx> writes:
In article <h6bglr$nvm$1@xxxxxxxxxxxxxxxxx>, kj <no.email@xxxxxxxxxxx>
wrote:
I know that it is possible for an N x N matrix to lack a set of N
linearly-independent eigenvectors. But I have no intuition at all
for such matrices. Is there a way to characterize them that would
enhance one's intuition? Importantly, is there some type of
situation, or application, where such matrices are common?
And for further reference, is there a compact name for such matrices,
that I could use in a Google search to learn more about them?
TIA!
kynn
Begin by studying the matrix
[0 1]
[0 0]
Figure out why it has no basis of eigenvectors.
Then find "Jordan normal form" in your linear algebra textbook.
By the way, you are talking about "repeated eigenvalues" not "repeated
eigenvectors"
A useful way of thinking about these matrices in the case where there is
only one eigenvalue is that they are of the form r I + N where r is the
eigenvalue, I the n x n identity matrix and N a nilpotent matrix, i.e. a
matrix such that N^n = 0. The eigenvectors are the nonzero members of the
null space of N.
One place such a matrix arises geometrically is as the matrix for a shear
transformation.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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