Exchange matrix explained

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]

$\begin J_2 &= \begin 0 & 1 \\ 1 & 0 \end \\[4pt] J_3 &= \begin 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end \\ &\quad \vdots \\[2pt] J_n &= \begin 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & \,_ \!\, ^ \! \dot\phantom & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \end\end$

Definition

If is an exchange matrix, then the elements of are $J_ = \begin 1, & i + j = n + 1 \\0, & i + j \ne n + 1\\\end$

Properties

Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,

\begin0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0\end\begin1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end =\begin7 & 8 & 9 \\4 & 5 & 6 \\1 & 2 & 3\end.

Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,

\begin1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end\begin0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0\end =\begin3 & 2 & 1 \\6 & 5 & 4 \\9 & 8 & 7\end.

Exchange matrices are symmetric; that is:

J_n^\mathsf = J_n.

For any integer :

J_n^k = \begin I & \text k \text \\[2pt] J_n & \text k \text \end In particular, is an involutory matrix; that is,

J_n^ = J_n.

The trace of is 1 if is odd and 0 if is even. In other words:

\operatorname(J_n) = n\bmod 2.

The determinant of is:

\det(J_n) = (-1)^\frac 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 is:

\det(\lambda I- J_n) = \begin \big[(\lambda+1)(\lambda-1)\big]^\frac & \text n \text \\[4pt] (\lambda-1)^\frac(\lambda+1)^\frac & \text n \text \end

The adjugate matrix of is:

\operatorname(J_n) = \sgn(\pi_n) J_n. (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.

Exchange matrix explained

Definition

Properties

Relationships

See also

Notes and References