Involutory matrix

Inmathematics,aninvolutory matrixis asquare matrixthat is its owninverse.That is, multiplication by the matrix${\mathbf {A}}_{n\times n}$is aninvolutionif and only if${\mathbf {A}}^{2}={\mathbf {I}},$where${\mathbf {I}}$is the$n\times n$identity matrix.Involutory matrices are allsquare rootsof the identity matrix. This is a consequence of the fact that anyinvertible matrixmultiplied by its inverse is the identity.

Examples

The$2\times 2$realmatrix${\begin{pmatrix}a&b\\c&-a\end{pmatrix}}$is involutory provided that$a^{2}+bc=1.$

ThePauli matricesin⁠$M(2,\mathbb {C} )$⁠are involutory: $${\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\end{aligned}}$$

One of the three classes ofelementary matrixis 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 asignature matrix,all of which are involutory.

Some simple examples of involutory matrices are shown below.

$${\begin{array}{cc}\mathbf {I} ={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}};&\mathbf {I} ^{-1}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\\\\\mathbf {R} ={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}};&\mathbf {R} ^{-1}={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}\\\\\mathbf {S} ={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}};&\mathbf {S} ^{-1}={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}}\\\end{array}}$$ where

• Iis the 3 × 3 identity matrix (which is trivially involutory);
• Ris the 3 × 3 identity matrix with a pair of interchanged rows;
• Sis asignature matrix.

Anyblock-diagonal matricesconstructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.

Symmetry

An involutory matrix which is alsosymmetricis anorthogonal matrix,and thus represents anisometry(alinear transformationwhich preservesEuclidean distance). Conversely every orthogonal involutory matrix is symmetric. As a special case of this, everyreflectionand 180°rotation matrixis involutory.

Properties

An involution isnon-defective,and eacheigenvalueequals$\pm 1$,so an involutiondiagonalizesto a signature matrix.

Anormalinvolution isHermitian(complex) or symmetric (real) and alsounitary(complex) or orthogonal (real).

Thedeterminantof an involutory matrix over anyfieldis ±1.

IfAis ann × nmatrix, thenAis involutory if and only if${\mathbf {P}}_{+}=({\mathbf {I}}+{\mathbf {A}})/2$isidempotent.This relation gives abijectionbetween involutory matrices and idempotent matrices.Similarly,Ais involutory if and only if${\mathbf {P}}_{-}=({\mathbf {I}}-{\mathbf {A}})/2$isidempotent.These two operators form the symmetric and antisymmetric projections$v_{\pm }={\mathbf {P}}_{\pm }v$of a vector$v=v_{+}+v_{-}$with respect to the involutionA,in the sense that${\mathbf {A}}v_{\pm }=\pm v_{\pm }$,or${\mathbf {AP}}_{\pm }=\pm {\mathbf {P}}_{\pm }$.The same construct applies to anyinvolutory function,such as thecomplex conjugate(real and imaginary parts),transpose(symmetric and antisymmetric matrices), andHermitian adjoint(Hermitianandskew-Hermitianmatrices).

IfAis an involutory matrix in⁠$M(n,\mathbb {R} ),$⁠which is amatrix algebraover thereal numbers,andAis not a scalar multiple ofI,then thesubalgebra$\{x{\mathbf {I}}+y{\mathbf {A}}:xy\in \mathbb {R} \}$generated byAisisomorphicto thesplit-complex numbers.

IfAandBare two involutory matrices whichcommutewith each other (i.e.AB=BA) thenABis also involutory.

IfAis an involutory matrix then everyintegerpowerofAis involutory. In fact,Aⁿwill be equal toAifnisoddandIifniseven.

See also

• Affine involution