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

Properties

• Exchange matrices are symmetric

• For any integer In particular, is an involutory matrix

• The trace of is 1 if is odd and 0 if

• The determinant of As a function of , it has period 4, giving 1, 1, −1, −1 when is congruent modulo 4 to 0, 1, 2, and 3 respectively.

• The characteristic polynomial of

its eigenvalues).

• The adjugate matrix of (where is the sign of the permutation of elements).

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

• Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)

References