How Does Permuting Eigenvectors Affect Matrix Representations?

[Cylab]

A = PD[P][/-1];
A: square matrix;
D: is a matrix of Jordan canonical form
P:is EigenVectors..(p1,p2,p3,p4...pr)

Is it possible to permute the sequence p1,p2,p3,p4...pr into other form?
I know it should be possible to permute...
How should permute it and what significance it has?
Why is it essential to permute vectors/matrix?
Thanks in advance for your attention.

 

[HallsofIvy]

If an n by n matrix is "diagonalizable", that means it has n independent eigenvectors and so if you use those eigenvectors as basis vectors will give a diagonal matrix representing the same linear transformation as the original matrix- the two matrices are "similar". Changing the order in which you use the eigenvectors will give a diagonal matrix with the number on the diagonal in different places, but they will still be similar to the original matrix and so to each other.

If a matrix is not diagonalizable, it does not have a "complete set" of eigenvectors- you can not have a basis for the vector space consisting entirely of eigenvetors. But you can, by using "generalized eigenvectors" (v is a "generalized eigenvector" corresponding to eigenvalue $\lambda$ if it is NOT true that $Av= \lambda v$ but it is true that Av is an eigenvalue or another generalized eigenvector.) The eigenvector and generalized eigenvectors corresponding to a given eigenvalue give the "Jordan" blocks in the Jordan Normal form. Changing the order of those will give a matrix with the rows and columns rearranged and so not in "Jordan Normal Form", but still "similar" to such a matrix.