Inmathematics, 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]
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;
IfA 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).