What are the criteria for determining if a matrix is diagonalizable?

[KD-jay]

Homework Statement

1) Let's say I was trying to find the eigenvalues of a matrix and came up with the following characteristic polynomial:
λ(λ-5)(λ+2)
This would yield λ=0,5,-2 as eigenvalues. I'm kinda thrown off as to what the algebraic multiplicity of the eigenvalue 0 would be? I'm pretty sure it would just be 1 but I think I've misheard my instructor say otherwise during one example.

2) Let's say I have the following characteristic polynomial of matrix A.
(λ-2)(λ-4)(λ-α), where α is any number.

If I were trying to figure out values of α that make A diagonalizable, it would be any values of alpha that makes that particular eigenvalue different from 2 or 4. What if λ-α=0 did make λ equal 2 or 4? Do I have to check the geometric multiplicities for λ=2,4, or can I automatically assume that the matrix would not be diagonalizable.

Homework Equations

det(A-λI) = 0 where A is a matrix and λ are the eigenvalues A

The Attempt at a Solution

 

[Dick]

Yes, the algebraic multiplicity of 0 would be 1. Just like 5 and -2. There's nothing special about 0. And for the second one if alpha=2 then you need to check if there are two linearly independent eigenvectors for the value 2. There might be and there might not be.