We’re concerned with calculating eigenvalues.
Which matrices have the same eigenvalues?
Definition. Two matrices and are similar if there is some invertible matrix such that .
If is similar to , then is similar to .
Theorem. If and are similar matrices, then and have the same characteristic equation, and thus have the same eigenvalues (with the same multiplicities).
Proof.
We need to show that , as this would mean they have the same characteristic equation.
QED.
Note: Just because and have the same eigenvalues doesn’t mean they have the same eigenvectors.
is the result of multiplying a square matrix by itself times. This turns out to have many useful applications, so we want a way of computing it for large values of that’s quicker than doing manual matrix multiplication.
If is a diagonal matrix, then is the matrix formed by raising the entries on the diagonal of to the th power. This is a lot easier to compute naively than powers of non-diagonal matrices.
Definition. A matrix is said to be diagonalizable if is similar to a diagonal matrix.
By induction, , where is a diagonalizable matrix and is the diagonal matrix it’s similar to.
Theorem: An matrix is diagonalizable if and only if has linearly independent eigenvectors.
Theorem: An matrix with distinct eigenvalues is diagonalizable. (This is from another theorem stating that eigenvectors that correspond to distinct eigenvalues are linearly independent.)
In order to determine if is diagonalizable, we can find the eigenvalues of by solving for , then determining whether or not they are distinct. If they are, then is definitely diagonalizable. Otherwise, we have to find eigenvectors for each eigenvalue and see if they are linearly independent. In this way, determining if a matrix is diagonalizable is now a completely algorithmic process.
Ex. Is this matrix diagonalizable?
Yes, because its eigenvalues are just the entries on its diagonal, which there are four of, and they are all distinct.
(Note that this matrix is non-invertible – non-invertible matrices can be diagonalizable.)