Involutory matrix

Template:Short description In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix ${𝐀}_{n\times n}$ is an involution if and only if ${𝐀}^2=𝐈,$ where $𝐈$ is the $n\times n$ 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.

Examples

The $2\times 2$ real matrix $\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$ is involutory provided that $a^2+bc=1.$

The Pauli matrices in Template:Tmath are involutory: $$\begin{array}{cc}{\sigma }_1={\sigma }_x& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\\ {\sigma }_2={\sigma }_y& =\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),\\ {\sigma }_3={\sigma }_z& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).\end{array}$$

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.

$$\begin{array}{cc}𝐈=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right);& {𝐈}^{-1}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\\ \\ 𝐑=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right);& {𝐑}^{-1}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)\\ \\ 𝐒=\left(\begin{array}{ccc}+1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right);& {𝐒}^{-1}=\left(\begin{array}{ccc}+1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)\end{array}$$ where

• Template:Math is the 3 × 3 identity matrix (which is trivially involutory);
• Template:Math is the 3 × 3 identity matrix with a pair of interchanged rows;
• Template:Math is a signature matrix.

Any block-diagonal matrices 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 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 $\pm 1$, 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 Template:Math is an Template:Math matrix, then Template:Math is involutory if and only if ${𝐏}_{+}=(𝐈+𝐀)/2$ is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices. Similarly, Template:Math is involutory if and only if ${𝐏}_{-}=(𝐈-𝐀)/2$ is idempotent. These two operators form the symmetric and antisymmetric projections $v_{\pm }={𝐏}_{\pm }v$ of a vector $v=v_{+}+v_{-}$ with respect to the involution Template:Math, in the sense that $𝐀v_{\pm }=\pm v_{\pm }$, or ${𝐀𝐏}_{\pm }=\pm {𝐏}_{\pm }$. 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 Template:Math is an involutory matrix in Template:Tmath which is a matrix algebra over the real numbers, and Template:Math is not a scalar multiple of Template:Math, then the subalgebra $\{x𝐈+y𝐀:xy\in ℝ\}$ generated by Template:Math is isomorphic to the split-complex numbers.

If Template:Math and Template:Math are two involutory matrices which commute with each other (i.e. Template:Math) then Template:Math is also involutory.

If Template:Math is an involutory matrix then every integer power of Template:Math is involutory. In fact, Template:Math will be equal to Template:Math if Template:Mvar is odd and Template:Math if Template:Mvar is even.

See also

• Affine involution