Involutory matrix

Symmetry

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. As a special case of this, every reflection and 180° rotation matrix 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.

If A is an n × n matrix, then A is involutory if and only if ${\displaystyle {\mathbf{P}}_{+}=(\mathbf{I}+\mathbf{A})/2}$ is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices. Similarly, A is involutory if and only if ${\displaystyle {\mathbf{P}}_{-}=(\mathbf{I}-\mathbf{A})/2}$ is idempotent. These two operators form the symmetric and antisymmetric projections ${\displaystyle v_{\pm }={\mathbf{P}}_{\pm }v}$ of a vector ${\displaystyle v=v_{+}+v_{-}}$ with respect to the involution A, in the sense that ${\displaystyle \mathbf{A}{v}_{\pm }=\pm v_{\pm }}$, or ${\displaystyle {\mathbf{A}\mathbf{P}}_{\pm }=\pm {\mathbf{P}}_{\pm }}$. 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 ⁠${\displaystyle M(n,\mathbb{R}),}$⁠ which is a matrix algebra over the real numbers, and A is not a scalar multiple of I, then the subalgebra ${\displaystyle \{x\mathbf{I}+y\mathbf{A}:xy\in \mathbb{R}\}}$ generated by A is isomorphic to the split-complex numbers.

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.