Diagonalization

Diagonalization is used all the time in computer graphics, physics, and engineering. It's a way of simplifying a transformation by changing the basis to one of eigenvectors. This is a powerful tool, and it's a great way to understand the power of eigenvectors and eigenvalues.

Diagonalizability
A square matrix is diagonalisable if it is similar to a diagonal matrix.

Thm. An matrix is diagonalisable if and only if it has linearly independent eigenvectors.

While we can't guarantee that is similar to a unique diagonal matrix, we can guarantee the following: If is similar to the diagonal matrices and , then and have the same set of diagonal elements (with the same multiplicities).

Thm. Let be a square matrix. For each positive integer , if is are eigenvectors of with distinct eigenvalues , then is linearly independent (this is just saying that eigenvectors with distinct eigenvalues are linearly independent).

Thm. If is and has distinct real eigenvalues, then is diagonalizable.

But what if doesn't have distinct real eigenvalues? It may still be diagonlizable:

Thm. If is an matrix with with real eigenvalues , and denotes the eigenspace of for each . Then, is diagonalizable if and only if Thm. If is an eigenvalue of the matrix , then
because is the kernel of .

What's the point of diagonalization?
Let's consider some matrix that represents a transformation. What if we want to apply it three times ()? Well, we could multiply by itself three times, but that's a lot of work.
If we can find a diagonal matrix that is similar to , then , and . But is just the diagonal elements cubed! So, we can find by just cubing the diagonal elements of .
Diagonalization also makes it easy to find the inverse of a matrix. If , then , and is just the reciprocal of the diagonal elements.

You'll find diagonalization everywhere you go. For instance, in probability theory, you'll see that the transition matrix of a Markov chain is diagonalizable. In physics, you'll see that the Hamiltonian operator is diagonalizable. In computer graphics, you'll see that the transformation matrix is diagonalizable. It's a powerful tool, and it's worth understanding.

~ The End ~

What I've covered here is just the beginning of linear algebra. There's so much more to learn, and I hope you continue to explore. If you have any questions, feel free to reach out to me. Good luck!