Today I Learned
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.