Few questions:

1. Are the eigenvectors of a diagonal matrix always orthogonal?

2. Consider a non-symmetric matrix. Diagonalize it. It undergoes a transformation into a diagonal matrix.

Assume that the original non-symmetric matrix has independent columns and so possesses linearly independent eigenvectors which however, are not mutually orthogonal.

Since the directions of the eigenvectors remain invariant under a transformation, are the eigenvectors of the resulting diagonal matrix non-orthogonal??

Can a geometric explanation be provided?

Diagonalization represents a change of basis to a basis of eigenvectors.
1. A matrix, that is diagonal in the basis unit vectors (1,0,0..0), (0,1,0..0) etc. has just these unit vectors as eigenvectors. And they are clearly orthogonal.



2. A non-symmetric n by n matrix of full rank (with indepedent columns) does not in general have a full rank n by n matrix of eigenvectors. Thus you ar not in general able to diagonalize it.

Anyway...
Changing to a basis, which is not orthogonal you can visualize as a warping of the entire space as to make the basis orthogonal. The resulting diagonal matrix of your example only has orthogonal columns in the sense that they are in the same directions as the basis of eigenvectors.

Change of bases is an important concept, but can seem difficult.

For a geometric approach to understanding eigenvalues and -vectors se lecture 20 of the OCW course on tensors http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-60Fall-2005/VideoLectures/index.htm

Keep in mind that a second rank tensor (rank meens something else here) can be represented simply by a matrix.  

--Last edited by zee on 2007-01-25 20:04:31 --

** Let V be a vector space and T: V→V be a linear transformation. A scalar is called an eigen-value for T if there exists a non-zero vector vV such that Tv =  λv. The non-zero vector  v is called an eigenvector for the eigenvalue λ .**


http://www.mathresource.iitb.ac.in/linear%20algebra/mainchapter9.4.html  

--Last edited by ferme on 2007-02-26 08:40:26 --