Involutory matrix

The 2\times2 real matrix \begin{pmatrix}a & b \\ c & -a \end{pmatrix} is involutory provided that a^2 + bc = 1 . The Pauli matrices in 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 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. \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 • is the 3 × 3 identity matrix (which is trivially involutory); • is the 3 × 3 identity matrix with a pair of interchanged rows; • is a signature matrix. Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks. == Symmetry ==

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric. As a special case of this, every reflection and 180° rotation matrix is involutory. ==Properties==

An involution is non-defective, and each eigenvalue equals \pm 1, so an involution diagonalizes to a signature matrix. A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real). The determinant of an involutory matrix over any field is ±1. If is an matrix, then is involutory if and only if \bold P_+ = (\bold I + \bold A)/2 is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices. Similarly, is involutory if and only if \bold P_- = (\bold I - \bold A)/2 is idempotent. These two operators form the symmetric and antisymmetric projections v_\pm = \bold P_\pm v of a vector v = v_+ + v_- with respect to the involution , in the sense that \bold Av_\pm = \pm v_\pm, or \bold{A P}_\pm = \pm \bold P_\pm. The same construct applies to any involutory function, such as the complex conjugate (real and imaginary parts), transpose (symmetric and antisymmetric matrices), and Hermitian adjoint (Hermitian and skew-Hermitian matrices). If is an involutory matrix in which is a matrix algebra over the real numbers, and is not a scalar multiple of , then the subalgebra \{x \bold I + y \bold A: x y \in \R\} generated by is isomorphic to the split-complex numbers. If and are two involutory matrices which commute with each other (i.e. ) then is also involutory. If is an involutory matrix then every integer power of is involutory. In fact, will be equal to if is odd and if is even. ==See also==