108 Matrix form of the damped harmonic oscillator

with corresponding eigenvectors

\begin{bmatrix}1\\ \lambda_{+}\end{bmatrix},\qquad\begin{bmatrix}1\\ \lambda_{-}\end{bmatrix}

so when $\lambda_{+}\neq\lambda_{-}$ , we introduce

S=\begin{bmatrix}1&1\\ \lambda_{+}&\lambda_{-}\end{bmatrix}

so that $S$ is invertible and

A=S\begin{bmatrix}\lambda_{+}&0\\ 0&\lambda_{-}\end{bmatrix}S^{-1}

and

\exp(tA)=S\begin{bmatrix}e^{t\lambda_{+}}&0\\ 0&e^{t\lambda_{-}}\end{bmatrix}S^{-1}