Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse. That is, matrix A is an involution if and only if A² = I. One of the three classes of elementary matrix is involutory, 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 involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.

Involutory 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 involutory if and only if ½(A + I) is idempotent.

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). A reflection matrix is an example of an involutory matrix.

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

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 an identity 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 A^T below:

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

is involutory.

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

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

See also

• Involution (mathematics)
• Idempotence
• Affine involution