Steven O. said:

Math is a hobby for me. I've been reading up on Eigenvectors and

Eigenvalues. It get the manipulations involved, but can't imagine the

applications -- and the books I have don't help. Can people provide a

few examples?

Specific examples, if possible -- not just, they are used in

electronics, or physics, or whatever, but rather, something like:

M is the matrix which describes such-and-such physical property or

transformation or process, its eigenvectors V correspond to such and

such property, and the eigenvalues of V and M indicate such-and-such.

Imagine P being the matrix of transition probabilities from

one state of a system to another of some system.

P_ij is the probability that the system goes from state i to

state j. The sum of each row is one:

sum( P_ij, j = 1...n ) = 1.

This is the transition matrix of a so-called Markov chain.

Under certain circumstances the infinite matrix product

limit converges such that

limit( P_ij^(n), n-->infinity) = p_j for all i,j.

where [ p_j, j=1...n ] is the limit vector of the probabilities

of the system being in the different states.

Here P_ij^(n) is the i,j element of the product matrix P^n,

with the transition probabilities from state i to state j after

n steps (as opposed to after 1 step as P_ij).

In stead of calculating the limit, one can try to find the

vector [ p_i ] of the probabilities of the initial states,

such that these probabilities are not influenced by the

evolution of the system, i.o.w. find the vector [ p_i ]

such that

sum( p_i * P_ij, i=1...n ) = p_j for all j,

i.o.w. find an eigenvector with eigenvalue 1 of the

transposed matrix P^t.

This eigenvector with probabilities of the initial system

being in the different states, does not change when the

sytem evolves.

Dirk Vdm