Involutory matrix

In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix $\bold A_{n \times n}$ is an involution if and only if $\bold A^2 = \bold I,$ where $\bold I$ is the $n \times n$ 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.{{citation

| last = Higham | first = Nicholas J. | authorlink = Nicholas Higham

| contribution = 6.11 Involutory Matrices

| doi = 10.1137/1.9780898717778

| isbn = 978-0-89871-646-7

| location = Philadelphia, PA

| mr = 2396439

| pages = 165–166

| publisher = Society for Industrial and Applied Mathematics (SIAM)

| title = Functions of Matrices: Theory and Computation

| url = https://books.google.com/books?id=2Wz_zVUEwPkC&pg=PA165

| year = 2008}}.

Examples

The $2\times2$ real matrix $\begin{pmatrix}a & b \\ c & -a \end{pmatrix}$ is involutory provided that $a^2 + bc = 1 .$ Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, 2nd edition, pp 12,13 Academic Press {{ISBN|0-12-435560-9}}

The Pauli matrices in {{tmath|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 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

{{math|I}} is the 3 × 3 identity matrix (which is trivially involutory);
{{math|R}} is the 3 × 3 identity matrix with a pair of interchanged rows;
{{math|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.{{citation

| last = Govaerts | first = Willy J. F.

| doi = 10.1137/1.9780898719543

| isbn = 0-89871-442-7

| location = Philadelphia, PA

| mr = 1736704

| page = 292

| publisher = Society for Industrial and Applied Mathematics (SIAM)

| title = Numerical methods for bifurcations of dynamical equilibria

| url = https://books.google.com/books?id=rqvYq19qwiwC&pg=PA292

| year = 2000}}.

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.{{citation

| last = Bernstein | first = Dennis S.

| contribution = 3.15 Facts on Involutory Matrices

| edition = 2nd

| isbn = 978-0-691-14039-1

| location = Princeton, NJ

| mr = 2513751

| pages = 230–231

| publisher = Princeton University Press

| title = Matrix Mathematics

| url = https://books.google.com/books?id=-c0NxJg4vHMC&pg=PA230

| year = 2009}}.

If {{math|A}} is an {{math|n × n}} matrix, then {{math|A}} 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, {{math|A}} 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 {{math|A}}, 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 {{math|A}} is an involutory matrix in {{tmath|M(n, \R),}} which is a matrix algebra over the real numbers, and {{math|A}} is not a scalar multiple of {{math|I}}, then the subalgebra $\{x \bold I + y \bold A: x y \in \R\}$ generated by {{math|A}} is isomorphic to the split-complex numbers.

If {{math|A}} and {{math|B}} are two involutory matrices which commute with each other (i.e. {{math|1=AB = BA}}) then {{math|AB}} is also involutory.

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

References

Category:Matrices (mathematics)

Involutory matrix

Examples

Symmetry

Properties

See also

References