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