Eigenstates of a very 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.

Modified: by ;

Lei Ma (2015). 'Eigensystem of A Special Matrix', Datumorphism, 02 April. Available at: https://datumorphism.leima.is/til/math/eigensystem-of-a-special-matrix/.

Current Ref:

• til/math/eigensystem-of-a-special-matrix.md