Involutory matrix

In

. That is, multiplication by the matrix ${\displaystyle A_{n\times n}}$ is an involution if and only if ${\displaystyle A^{2}=I}$, where ${\displaystyle I}$ is the ${\displaystyle n\times n}$ identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.^[1]

Examples

The ${\displaystyle 2\times 2}$ real matrix ${\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}}$ is involutory provided that ${\displaystyle a^{2}+bc=1.}$^[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 3 × 3 identity matrix (which is trivially involutory);
• R is the 3 × 3 identity matrix with a pair of interchanged rows;
• S is a signature matrix.

Any

constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.

Symmetry

An involutory matrix which is also

As a special case of this, every is involutory.

Properties

An involution is non-defective, and each eigenvalue equals ${\displaystyle \pm 1}$, so an involution diagonalizes to a signature matrix.

A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real).

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 P₊ = (I + A)/2 is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.^[4] Similarly, A is involutory if and only if P₋ = (I − A)/2 is idempotent. These two operators form the symmetric and antisymmetric projections ${\displaystyle v_{\pm }=P_{\pm }v}$ of a vector ${\displaystyle v=v_{+}+v_{-}}$ with respect to the involution A, in the sense that ${\displaystyle Av_{\pm }=\pm v_{\pm }}$, or ${\displaystyle AP_{\pm }=\pm P_{\pm }}$. The same construct applies to any involutory function, such as the complex conjugate (real and imaginary parts), transpose (symmetric and antisymmetric matrices), and Hermitian adjoint (Hermitian and skew-Hermitian matrices).

If A is an involutory matrix in M(n, R), which is a

.

If A and B are two involutory matrices which commute with each other (i.e. AB = BA) then AB is also involutory.

If A is an involutory matrix then every integer power of A is involutory. In fact, Aⁿ will be equal to A if n is odd and I if n is even.

See also

• Affine involution

References