My textbook gives two theorems that are used to determine if a matrix is diagonilable
1)An n x n matrix is diagonalizable if and only if it has n linearly independent eigenvectors
2)If the roots of the characteristic polynomial of an n x n matrix A are all distinct then A is diagonizable
So if I have characteristic polynomial with two roots that are the same (e.g. but it has n linearly independent eigenvectors, which one takes presidents? Or should the two always agree and this means I made a mistake?
The problem that can occur is if you have a repeated eigenvalue say of multiplicity 2 the eigenspace for that eigenvalue may only have 1 linearly indpendant eigenvector. So the matrix is not diagonalizable. But if you do get two linearly independant eigenvectors then the matrix would be diagonalizable.