Involutary matrix

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In mathematics, an involutary matrix is a matrix that is its own inverse. That is, matrix A is an involution if A2 = I. One of the three classes of elementary matrix is involutary, namely the row-interchange elementary matrix. A special case of another class of elementary matrix, that which represents multiplication of a row or column by −1, is also involutary; it is in fact a trivial example of a signature matrix, all of which are involutary.

Involutary matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity. If A is an n×n matrix, then A is involutary if ½(A+I) is idempotent.

An involutary matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance).

Clearly, any block-diagonal matrices constructed from involutary matrices will also be involutary, as a consequence of the linear independence of the blocks.

[edit] Examples

Some simple examples of involutary matrices are shown below.

\begin{array}{cc} I=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} ; & I^{-1}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ \\ R=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} ; & R^{-1}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\ \\ S=\begin{pmatrix} +1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} ; & 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 by 2 matrices, where any matrix that may be written in the form A or AT below:

A=\begin{pmatrix} a & b \\ \frac{(1-a^2)}{b} & -a \end{pmatrix};\quad A^T=\begin{pmatrix} a & \frac{(1-a^2)}{b} \\ b & -a \end{pmatrix}

is involutary.

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

M=\begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix};\quad\Longrightarrow\quad 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} = I

[edit] See also

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