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
- 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.
- Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)