Exchange matrix
Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere
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]
Definition
If J is an n × n exchange matrix, then the elements of J are
Properties
- 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).
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.
- Symmetric matrices A that satisfy the condition AJ = JA are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.
See also
- Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)
References
- ^ Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.
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