Involutory matrix
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix is an involution if and only if , where is the 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.[1]
Examples
[edit]The real matrix is involutory provided that [2]
The Pauli matrices in M(2, C) are involutory:
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.
where
- I is the 3 × 3 identity matrix (which is trivially involutory);
- R is the 3 × 3 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
[edit]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 and 180° rotation matrix is involutory.
Properties
[edit]An involution is non-defective, and each eigenvalue equals , 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.[4]
If A is an n × n matrix, then A is involutory if and only if P+ = (I + A)/2 is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.[4] Similarly, A is involutory if and only if P− = (I − A)/2 is idempotent. These two operators form the symmetric and antisymmetric projections of a vector with respect to the involution A, in the sense that , or . 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 A is an involutory matrix in M(n, R), which is a matrix algebra over the real numbers, and A is not a scalar multiple of I, 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 (i.e. AB = BA) 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
[edit]References
[edit]- ^ 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.
- ^ Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, 2nd edition, pp 12,13 Academic Press ISBN 0-12-435560-9
- ^ 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.
- ^ a b 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.