Re: significane of eigenvalues and singular values


In article <1163579543.645273.302530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"paluri" <santoshpaluri@xxxxxxxxx> writes:
Hi all,

I am a novice in linear algebra...and have started working on it. I
have some basic dobuts:

1. what is the physical significane of eigenvectors and eigenvalues. I
have referred textbooks but everywhere only the numerical methods to
solve them are given. I want to know what important information does
the eignvalues and eigenvectors reveal about the matrix. Can i know
some good example to get the physical usage of eigenvectors. for
example i tried to study the markov matrix computation which is done by
a search engine and then it tries to compute the principal eigenvector
, i.e. with largest eigenvalue..but i didnt understand the reason
behind it.

there is no unique "physical" meaning of eigenvalues/eigenvectors
they appear e.g. in the analysis of vibrations: then they mean frequencies
and vibration modes
in your example of markov chain the eigenvector to the (dominant)
eigenvalue 1 means the stationary distribution
in the analysis of ordinary differential equations they describe stability/
instability of a process
in the analysis of iteration processes the eigenvalues describe the
damping of errors
etc



2. what is the exact difference and significane of a singular value and
an eigenvalue of a matrix. As it is not possible to calculate the
eigenvalue of a non square matrix whereas a singular value can
computed..!

I would be thankful for any suggestion!

if
A is hermitian (hence also square) and positive definite then
singular value = eigenvalue
if
A is hermitian then
singular value = abs(eigenvalue)
if A is general, whether square or not, then there is _no_, really _no_
relation between the two:
singular value =root of eigenvalue of (hermitian(A)*A) or zero

hth
peter
.

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