# Today I Learned

## Eigensystem of A Special Matrix

One of the most used matrix in physics is

where $a$, $b$, $c$ are real numbers.

It is interesting that as we go from

to the previous matrix, the eigenstates change from

to

no matter how small $b$ is.

**A useful trick when solving the eigensystem is to remove an identity from the matrix because it only shifts the eigenvalue by a certain amount.**