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Exchange matrix

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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


  • 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:
  • For any integer k: In particular, Jn is an involutory matrix; that is,
  • The trace of Jn is 1 if n is odd and 0 if n is even. In other words:
  • The determinant of Jn is: 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:
  • The adjugate matrix of Jn is: (where sgn is the sign of the permutation πk of k elements).


  • 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.