Exchange matrix
Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere From Wikipedia, the free encyclopedia
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:
its eigenvalues are 1 (with multiplicity ) and -1 (with multiplicity ).
- 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
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