In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A n × n {\displaystyle {\mathbf {A}}_{n\times n}} is an involution if and only if A 2 = I , {\displaystyle {\mathbf {A}}^{2}={\mathbf {I}},} where I {\displaystyle {\mathbf {I}}} is the n × n {\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.
Examples The 2 × 2 {\displaystyle 2\times 2} real matrix ( a b c − a ) {\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}} is involutory provided that a 2 + b c = 1. {\displaystyle a^{2}+bc=1.}
The Pauli matrices in M ( 2 , C ) {\displaystyle M(2,\mathbb {C} )} are involutory:
σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) . {\displaystyle {\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 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.
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