What Properties of a Matrix Can Eigenvalues and Eigenvectors Reveal?

[matqkks]

We are aware that by knowing the eigenvalues and eigenvectors we can evaluate the determinant, say if it is invertible and diagonalize to find powers of matrices.
Is there a list of properites of a matrix we can find by eigenvalues and eigenvectors?
Are there things that e.values and e.vectors cannot tell us about the matrix?

 

[M Quack]

You can also find the trace, which is the sum of all Eigenvalues.

I believe that you can construct the entire matrix from the Eigenvalues and Eigenvectors, but I can't remember the exact formula off the top of my head. You construct a diagonal matrix with the Eigenvectors on the diagonal, and a matrix composed of all the Eigenvectors.

It goes something like this:

Say $v_i$ is the Eigenvector with Eigenvalue $\lambda_i$

$M \cdot v_i = \lambda_i v_i$,

Define the diagonal matrix with the Eigenvalues

$D_{ij} = \delta_{ij} \lambda_i$

and a matrix composed of all the Eigenvectors

$V_{ij} = (v_i)_j$

Then you should get
$(V \cdot D \cdot V^T) \cdot v_i = \lambda_i v_i$

We have therefore reconstructed the original matrix
$M = V \cdot D \cdot V^T$

(Somebody please check, I'm making this up as we go along)

Since you can construct the original matrix from the Eigenvectors and Eigenvalues, you can determine each and every property of the original matrix.

 

[M Quack]

Eigenvectors have to be orthonormal for this to work, btw.