Exchange matrix

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.

$$\begin{array}{cc}{J}_2& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\\ J_3& =\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\\ & ⋮\\ J_n& =\left(\begin{array}{ccccc}0& 0& \cdots & 0& 1\\ 0& 0& \cdots & 1& 0\\ ⋮& ⋮& {}_{{}_{\cdot }}{}^{{}_{{}_{\cdot }}}\dot{\phantom{j}}& ⋮& ⋮\\ 0& 1& \cdots & 0& 0\\ 1& 0& \cdots & 0& 0\end{array}\right)\end{array}$$

Definition

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

Properties

• Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,$$\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)=\left(\begin{array}{ccc}7& 8& 9\\ 4& 5& 6\\ 1& 2& 3\end{array}\right).$$
• Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,$$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}3& 2& 1\\ 6& 5& 4\\ 9& 8& 7\end{array}\right).$$
• Exchange matrices are symmetric; that is: $$J_n^{𝖳}=J_n.$$
• For any integer k: $$J_n^k=\{\begin{array}{cc}I& \text{ if }k\text{ is even,}\\ J_n& \text{ if }k\text{ is odd.}\end{array}$$In particular, J_n is an involutory matrix; that is, $$J_n^{-1}=J_n.$$
• The trace of J_n is 1 if n is odd and 0 if n is even. In other words: $$\mathrm{tr}(J_n)=\frac{1-(-1{)}^n}{2}=nmod2.$$
• The determinant of J_n is: $$\det (J_n)=(-1{)}^{\lfloor n/2\rfloor }=(-1{)}^{\frac{n(n-1)}{2}}$$ 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 J_n is: $$\det (\lambda I-J_n)=(\lambda -1{)}^{\lceil n/2\rceil }(\lambda +1{)}^{\lfloor n/2\rfloor }=\{\begin{array}{cc}[(\lambda +1)(\lambda -1){]}^{\frac{n}{2}}& \text{ if }n\text{ is even,}\\ (\lambda -1{)}^{\frac{n+1}{2}}(\lambda +1{)}^{\frac{n-1}{2}}& \text{ if }n\text{ is odd,}\end{array}$$

its eigenvalues are 1 (with multiplicity $\lceil n/2\rceil$) and -1 (with multiplicity $\lfloor n/2\rfloor$).

• The adjugate matrix of J_n is: $$\mathrm{adj}(J_n)=\mathrm{sgn}({\pi }_n)J_n.$$ (where sgn is the sign of the permutation π_k 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 = JA^T 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)