Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A² = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity.[1]

Examples

The 2 × 2 real matrix ${\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}}$ is involutory provided that ${\displaystyle a^{2}+bc=1.}$[2]

The Pauli matrices in M(2,C) are involutory:

${\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,.\end{aligned}}}$

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.

${\displaystyle {\begin{array}{cc}\mathbf {I} ={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}};&\mathbf {I} ^{-1}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\\\\\mathbf {R} ={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}};&\mathbf {R} ^{-1}={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}\\\\\mathbf {S} ={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}};&\mathbf {S} ^{-1}={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}}\\\end{array}}}$

where

I is the identity matrix (which is trivially involutory);

R is an identity matrix with a pair of interchanged rows;

S 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.[3] As a special case of this, every reflection matrix is an involutory.

Properties

The determinant of an involutory matrix over any field is ±1.[4]

If A is an n × n matrix, then A is involutory if and only if ½(A + I) is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.[4]

If A is an involutory matrix in M(n, ℝ), a matrix algebra over the real numbers, then the subalgebra {x I + y A: x,y ∈ ℝ} generated by A is isomorphic to the split-complex numbers.

If A and B are two involutory matrices which commute with each other then AB is also involutory.

If A is an involutory matrix then every integer power of A is involutory. In fact, Aⁿ will be equal to A if n is odd and I if n is even.

See also

• Affine involution

References

1. Higham, Nicholas J. (2008), "6.11 Involutory Matrices", Functions of Matrices: Theory and Computation, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), pp. 165–166, doi:10.1137/1.9780898717778, ISBN 978-0-89871-646-7, MR 2396439.
2. Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, 2nd edition, pp 12,13 Academic Press ISBN 0-12-435560-9
3. Govaerts, Willy J. F. (2000), Numerical methods for bifurcations of dynamical equilibria, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), p. 292, doi:10.1137/1.9780898719543, ISBN 0-89871-442-7, MR 1736704.
4. Bernstein, Dennis S. (2009), "3.15 Facts on Involutory Matrices", Matrix Mathematics (2nd ed.), Princeton, NJ: Princeton University Press, pp. 230–231, ISBN 978-0-691-14039-1, MR 2513751.