Involutory Matrix - Examples

Examples

Some simple examples of involutory matrices are shown below.

\begin{array}{cc} \mathbf{I}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} ; & \mathbf{I}^{-1}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ \\ \mathbf{R}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} ; & \mathbf{R}^{-1}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\ \\ \mathbf{S}=\begin{pmatrix} +1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} ; & \mathbf{S}^{-1}=\begin{pmatrix} +1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \\ \end{array}

where

I is the identity matrix (which is trivially involutory);
R is a matrix with a pair of interchanged rows;
S is a signature matrix.

An interesting general condition exists, for 2 × 2 matrices, where any matrix that may be written in the form A or AT below:

is involutory.

For example, for a matrix M of this form, where we set a = 1, b = 1, we have

\mathbf{M}=\begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix};\quad\Longrightarrow\quad \mathbf{M}^2=\begin{pmatrix} 1\times 1+1\times 0 & 1\times 1+1\times -1 \\ 0\times 1-1\times 0 & 0\times 1-1\times -1 \end{pmatrix} =\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \mathbf{I}

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