Exchange matrix

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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.[1]


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.
  • Symmetric matrices A that satisfy the condition AJ = JA are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.

See also[edit]

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


  1. ^ Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.