Register to reply

Motivations for eigenvalues/vectors

by eckiller
Tags: motivations
Share this thread:
May1-05, 05:00 PM
P: 45

I understand one of the motivations for eigenvalues/vectors is when you need
to compute A^k * x. So we like to write,

A = C*D*C^-1 and then A^k = C * D^k * C^-1, and D^k is trivial to compute.

My professor said C^-1 and C can be though of as change of coordinate
matrices. But from which basis? For example, C^-1 would take me from
*some* basis to the basis of eigenvectors. But what is this *some* basis?

Is it assumed that everything is coordinitized relative to some basis B in
R^n. And then I want to change to the basis of eigenvectors B'?
Phys.Org News Partner Science news on
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
May1-05, 05:06 PM
P: 998
Short answer: Yes.

A is the matrix of a transformation wrt some basis B. D is the matrix of the same transformation wrt the eigenbasis B'. [itex]C^{-1}[/itex] takes vectors from B to B', and C takes vectors back from B' to B.
May1-05, 06:58 PM
Sci Advisor
PF Gold
P: 39,310
A general linear transformation can be written as a matrix in a given basis. If all you are given is the matrix, then the corresponding "given basis" is <1, 0,...>, <0,1,...> , etc. The basis you are changing to is the basis consisting of the eigenvectors for the matrix A.

(A matrix can be diagonalized if and only if there exist a basis consisting entirely of eigenvectors of the matrix.)

May1-05, 09:00 PM
P: 45
Motivations for eigenvalues/vectors

Thanks for clearing that up. I wish both of my linear algebra textbooks made it clear which basis we were in.
matt grime
May2-05, 08:00 AM
Sci Advisor
HW Helper
P: 9,398
But it obviously goes from whatever basis A is written with respect to, and it doesn't matter what that basis is, which is why the book didn't state it.

Register to reply

Related Discussions
Eigenvectors/values and diff equations Calculus & Beyond Homework 3
Eigenvalue proof help Calculus & Beyond Homework 5
General how to find the eigenvectors/values Linear & Abstract Algebra 4
Binomial Theorem - small values of x and approximate values Precalculus Mathematics Homework 7
Verify that a vector is an eigenvector of a matrix Introductory Physics Homework 4