Inmathematics, aninvolutory matrix is asquare matrix that is its owninverse. That is, multiplication by the matrix${\displaystyle {\mathbf{A}}_{n\times n}}$ is aninvolution if and only if${\displaystyle {\mathbf{A}}^2=\mathbf{I},}$ where${\displaystyle \mathbf{I}}$ is the${\displaystyle n\times n}$identity matrix. Involutory matrices are allsquare roots of the identity matrix. This is a consequence of the fact that anyinvertible matrix multiplied by its inverse is the identity.^[1]

The${\displaystyle 2\times 2}$real matrix is involutory provided that${\displaystyle a^2+bc=1.}$^[2]

ThePauli matrices in⁠${\displaystyle M(2,\mathbb{C})}$⁠ are involutory:

One of the three classes ofelementary 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 asignature 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 asignature matrix.

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

An involutory matrix which is alsosymmetric is anorthogonal matrix, and thus represents anisometry (alinear transformation which preservesEuclidean distance). Conversely every orthogonal involutory matrix is symmetric.^[3]As a special case of this, everyreflection and 180°rotation matrix is involutory.

An involution isnon-defective, and eacheigenvalue equals${\displaystyle \pm 1}$, so an involutiondiagonalizes to a signature matrix.

Anormal involution isHermitian (complex) or symmetric (real) and alsounitary (complex) or orthogonal (real).

Thedeterminant of an involutory matrix over anyfield is ±1.^[4]

IfA is ann × n matrix, thenA is involutory if and only if${\displaystyle {\mathbf{P}}_{+}=(\mathbf{I}+\mathbf{A})/2}$ isidempotent. This relation gives abijection between involutory matrices and idempotent matrices.^[4] Similarly,A is involutory if and only if${\displaystyle {\mathbf{P}}_{-}=(\mathbf{I}-\mathbf{A})/2}$ isidempotent. 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 involutionA, 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 anyinvolutory function, such as thecomplex conjugate (real and imaginary parts),transpose (symmetric and antisymmetric matrices), andHermitian adjoint (Hermitian andskew-Hermitian matrices).

IfA is an involutory matrix in⁠${\displaystyle M(n,\mathbb{R}),}$⁠ which is amatrix algebra over thereal numbers, andA is not a scalar multiple ofI, then thesubalgebra${\displaystyle \{x\mathbf{I}+y\mathbf{A}:xy\in \mathbb{R}\}}$generated byA isisomorphic to thesplit-complex numbers.

IfA andB are two involutory matrices whichcommute with each other (i.e.AB =BA) thenAB is also involutory.

IfA is an involutory matrix then everyintegerpower ofA is involutory. In fact,Aⁿ will be equal toA ifn isodd andI ifn iseven.

• Affine involution

• Matrices

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