Eigensystem of A Special Matrix

One of the most used matrix in physics is

$$ \begin{pmatrix} a + c \mathrm i & b \\ b & a + c \mathrm i \end{pmatrix}, $$

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

It is interesting that as we go from

$$ \begin{pmatrix} a + c \mathrm i & 0 \\ 0 & a + c \mathrm i \end{pmatrix}, $$

to the previous matrix, the eigenstates change from

$$ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \mathrm{and } \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$

to

$$ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \mathrm{and } \begin{pmatrix} 1 \\ -1 \end{pmatrix} $$

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.