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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

Official source

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