Exchange matrix

In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix. If J is an n × n exchange matrix, then the elements of J are Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e., Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e., Exchange matrices are symmetric; that is, JnT = Jn. For any integer k, Jnk = I if k is even and Jnk = Jn if k is odd. In particular, Jn is an involutory matrix; that is, Jn−1 = Jn. The trace of Jn is 1 if n is odd and 0 if n is even. In other words, the trace of Jn equals . The determinant of Jn equals . As a function of n, it has period 4, giving 1, 1, −1, −1 when n is congruent modulo 4 to 0, 1, 2, and 3 respectively. The characteristic polynomial of Jn is when n is even, and when n is odd. The adjugate matrix of Jn is . An exchange matrix is the simplest anti-diagonal matrix. Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric. Any matrix A satisfying the condition AJ = JAT is said to be persymmetric. Symmetric matrices A that satisfy the condition AJ = JA are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.