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.
Definition
If is an exchange matrix, then the elements of 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 : In particular, is an involutory matrix; that is,
- The trace of is 1 if is odd and 0 if is even. In other words:
- The determinant of is: As a function of , it has period 4, giving 1, 1, âÂÂ1, âÂÂ1 when is congruent modulo 4 to 0, 1, 2, and 3 respectively.
- The characteristic polynomial of is:
its eigenvalues are 1 (with multiplicity ) and -1 (with multiplicity ).
Relationships
- An exchange matrix is the simplest anti-diagonal matrix.
- Any matrix satisfying the condition is said to be centrosymmetric.
- Any matrix satisfying the condition is said to be persymmetric.
- Symmetric matrices that satisfy the condition are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.
See also
References