Conditions For Diagonalization
To check if a matrix is diagonalizable, there are two conditions:
For an $n \times n$ matrix, the matrix must have $n$ linearly independent Eigenvectors
- This is always true if $A$ has $n$ distinct Eigenvalues
More generally, $a.m.(\lambda) = g.m.(\lambda)$ for all eigenvalues of $A$.
- As a proof, think about this, $g.m(\lambda)$ tells us how many linearly independent eigenvectors exist for an eigenvalue $\lambda$ (due to the definition of Dimension). Moreover, since the sum of all the algebraic multiplicities must equal $n$, it is clear that when the algebraic multiplicity is equal to geometric multiplicity for all eigenvalues, we have $n$ linearly independent eigenvectors.