Rahul Sharma (Editor)

Exchange matrix

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

J 2 = ( 0 1 1 0 ) ; J 3 = ( 0 0 1 0 1 0 1 0 0 ) ; J n = ( 0 0 ⋯ 0 0 1 0 0 ⋯ 0 1 0 0 0 ⋯ 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 ⋯ 0 0 0 1 0 ⋯ 0 0 0 ) .

Explaining the meaning of an identity matric and an exchange matrix linear algebra2 1 15


Definition

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

J i , j = { 1 , j = n − i + 1 0 , j ≠ n − i + 1

Properties

  • 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.

    • Relationships

  • 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.

    • References

    Exchange matrix Wikipedia
    (Text) CC BY-SA


    Similar Topics