# Involutory matrix

In mathematics, an **involutory matrix** is a matrix that is its own inverse. That is, multiplication by matrix **A** is an involution if and only if **A**^{2} = **I**. 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.^{[1]}

## Examples[edit]

The 2 × 2 real matrix is involutory provided that ^{[2]}

The Pauli matrices in M(2,C) are involutory:

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.

Some simple examples of involutory matrices are shown below.

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.

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

## Symmetry[edit]

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric;^{[3]}
as a special case of this, every reflection matrix is an involutory.

## Properties[edit]

The determinant of an involutory matrix over any field is ±1.^{[4]}

If **A** is an *n × n* matrix, then **A** is involutory if and only if ½(**A** + **I**) is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.^{[4]}

If **A** is an involutory matrix in M(*n*, ℝ), a matrix algebra over the real numbers, then the subalgebra {*x* **I** + *y* **A**: *x,y* ∈ ℝ} generated by **A** is isomorphic to the split-complex numbers.

if **A** and **B** are two involutory matrices which commute with each other then **AB** is also involutory.

if **A** is involutory matrix then every natural power of **A** is involutory. In fact, **A**^{n} will be equal to **A** if *n* is odd and **I** if *n* is even.

## See also[edit]

## References[edit]

**^**Higham, Nicholas J. (2008), "6.11 Involutory Matrices",*Functions of Matrices: Theory and Computation*, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), pp. 165–166, doi:10.1137/1.9780898717778, ISBN 978-0-89871-646-7, MR 2396439.**^**Peter Lancaster & Miron Tismenetsky (1985)*The Theory of Matrices*, 2nd edition, pp 12,13 Academic Press ISBN 0-12-435560-9**^**Govaerts, Willy J. F. (2000),*Numerical methods for bifurcations of dynamical equilibria*, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), p. 292, doi:10.1137/1.9780898719543, ISBN 0-89871-442-7, MR 1736704.- ^
^{a}^{b}Bernstein, Dennis S. (2009), "3.15 Facts on Involutory Matrices",*Matrix Mathematics*(2nd ed.), Princeton, NJ: Princeton University Press, pp. 230–231, ISBN 978-0-691-14039-1, MR 2513751.