Exchange matrix

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

In mathematics, especially linear algebra, the exchange matrix (also called the reversal matrix, backward identity, or standard involutory permutation) is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.[1]

Definition[edit]

If J is an n×n exchange matrix, then the elements of J are defined such that:

Properties[edit]

  • JT = J.
  • Jn = I for even n; Jn = J for odd n, where n is any integer. Thus J is an involutory matrix; that is, J−1 = J.
  • The trace of J is 1 if n is odd, and 0 if n is even.
  • The characteristic polynomial is for even, and for odd.

Relationships[edit]

References[edit]

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