Matrix exponentials

The exponential of a matrix is defined as \[\begin{equation} \require{physics} \exp(\vb{A}) = \sum_{n=0}^{+\infty} \frac{\vb{A}^n}{n!} \thinspace . \end{equation}\] Important properties of matrix exponentials: \[\begin{align} & \det(\exp(\vb{A})) = \exp(\tr(\vb{A})) \\ & \exp(\vb{A})^\dagger = \exp(\vb{A}^\dagger) \\ & \vb{B} \thinspace \exp(\vb{A}) \thinspace \vb{B}^{-1} = \exp(\vb{B} \vb{A} \vb{B}^{-1}) \\ & \comm{\vb{A}}{\vb{B}} = 0 \implies \exp(\vb{A} + \vb{B}) = \exp(\vb{A}) \exp(\vb{B}) \\ & \exp(\vb{A}) \thinspace \vb{B} \thinspace \exp(-\vb{A}) = \vb{B} + \comm{\vb{A}}{\vb{B}} + \frac{1}{2!} \comm{\vb{A}}{\comm{\vb{A}}{\vb{B}}} \thinspace . \end{align}\] A unitary matrix \(\vb{U}\) can always be written as the exponential of an anti-Hermitian matrix \(\vb{X}\): \[\begin{equation} \vb{U} = \exp(\vb{X}) \qquad \vb{X}^\dagger = - \vb{X} \thinspace . \end{equation}\] A special unitary matrix \(\vb{O}\) is a unitary matrix with determinant equal to \(1\) and it can be written as \[\begin{equation} \vb{O} = \exp({}^R \vb{X} + i {}^I \vb{X}) \thinspace , \end{equation}\] where \({}^R \vb{X}\) is a real antisymmetric matrix: \[\begin{equation} {}^R \vb{X} = \frac{\vb{X} - \vb{X}^T}{2} \thinspace , \end{equation}\] and \({}^I \vb{X}\) is a real symmetric matrix with vanishing diagonal elements: \[\begin{equation} {}^I \vb{X} = \frac{\vb{X} + \vb{X}^T}{2i} \thinspace . \end{equation}\]

A special orthogonal matrix can be written as \[\begin{equation} \vb{R} = \exp({}^R \vb{X}) \qquad {}^R \vb{X}^T = - {}^R \vb{X} \end{equation}\] where \({}^R \vb{X}\) is a real antisymmetric matrix. It can be diagonalized, yielding \[\begin{equation} {}^R \vb{X} = i \vb{V} \thinspace \boldsymbol{\tau} \thinspace \vb{V}^\dagger \qquad \vb{V} \vb{V}^\dagger = \vb{I} \thinspace . \end{equation}\] \({}^R \vb{X}^2\) can also be diagonalized: \[\begin{equation} {}^R \vb{X}^2 = - \vb{W} \boldsymbol{\tau}^2 \vb{W}^T \qquad \vb{W} \vb{W}^T = \vb{I} \thinspace , \end{equation}\] such that a special orthogonal matrix can be calculated using real arithmetic as \[\begin{equation} \vb{R} = \vb{W} \thinspace \cos \boldsymbol{\tau} \thinspace \vb{W}^T + \vb{W} \thinspace \boldsymbol{\tau}^{-1} \sin \boldsymbol{\tau} \thinspace \vb{W}^T {}^R \vb{X} \thinspace . \end{equation}\]