EIGEN … THINGS (values, vectors, spaces … )

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Presentation transcript:

EIGEN … THINGS (values, vectors, spaces … ) (From Wikipedia)

CONVENTION: From now on, unless otherwise spec-ified, all matrices shall be square, i.e.

Another, less simple example:

What are these eigenthings good for? How many eigenvectors correspond to a fixed eigenvalue? How many eigenvalues correspond to a fixed eigenvector? How do we find eigenvalues? How do we find eigenvectors? How many eigenvalues are there? How many eigenvectors are there? We’ll answer them one at a time. Answer to no. 1. Just wait and have faith in math!

How many eigenvectors correspond to a fixed eigenvalue How many eigenvectors correspond to a fixed eigenvalue? Here is an eigenvector x :

How many eigenvalues correspond to a fixed eigenvector? How many eigenvalues are there? (Later) How many eigenvectors are there? This is easy, infinitely many or How do we find eigenvalues? How do we find eigenvectors? We will answer both at the same time.

Let me start with the trivial observation that, if we know that a particular scalar

corresponding eigenspace.

(Let’s check it out.) The example actually shows us how to find eigen-values (and then eigenspaces). Let

has a non-zero solution!

We have learned that in order to find all the eigen-values of an

Another easy theorem. If an

One last important application:

Theorem.