Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse. That is, matrix A is an involution iff 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 a 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.

:

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 A^T below:

:

is involutory.

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

:

ee also

*Involution (mathematics)
*Idempotence
*Affine involution