Concept

# Involutory matrix

Summary
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any invertible matrix multiplied by its inverse is the identity. Examples The 2 × 2 real matrix \begin{pmatrix}a & b \ c & -a \end{pmatrix} is involutory provided that a^2 + bc = 1 . The Pauli matrices in M(2, C) are involutory: \begin{align} \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{align} One of the three classes of elementary
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