How Can Eigenvalues and Eigenvectors Be Explained Geometrically?

[bPawn]

Hi,

I am doing my PhD in Informatics and especially graph theory. I came across eigen-analysis numerous times in the context of spectral graph theory.

The question:
I would like to see an *elegant geometric explanation* of the eigenvalues and eigenvectors of a Matrix A (symmetric, real, >= 0). (Suppose that the only thing I know about linear algebra is how to change between bases and do a few matrix operations)

Of course I have looked many textbooks, but they only write math, not essence.

thanks!

 

[wofsy]

Hi,

I am doing my PhD in Informatics and especially graph theory. I came across eigen-analysis numerous times in the context of spectral graph theory.

The question:
I would like to see an *elegant geometric explanation* of the eigenvalues and eigenvectors of a Matrix A (symmetric, real, >= 0). (Suppose that the only thing I know about linear algebra is how to change between bases and do a few matrix operations)

Of course I have looked many textbooks, but they only write math, not essence.

thanks!

Perhaps you can explain your question more. I am sure you know about the picture of the eigenvectors as principal axes of an ellipsoid and they can be found through sequential maximization of the inner product that the symmetric matrix determines.