mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of

permutation matrices In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...

, where the 1 elements reside on the

antidiagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...

and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

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.,

\begin
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end
\begin
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end =
\begin
7 & 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.,

\begin
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end
\begin
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end =
\begin
3 & 2 & 1 \\
6 & 5 & 4 \\
9 & 8 & 7
\end.

* Exchange matrices are

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

; that is:

J_n^\mathsf = J_n.

* For any

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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) = \frac = n\bmod 2.

* The

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

of is:

\det(J_n) = (-1)^ = (-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 In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

of is:

(\lambda-1)^\frac(\lambda+1)^\frac & \text n \text \end

its

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

are 1 (with multiplicity

\lceil n/2\rceil

) and -1 (with multiplicity

\lfloor n/2\rfloor

). * The adjugate matrix of is:

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

(where is the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

of the

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

of elements).

Relationships

* An exchange matrix is the simplest anti-diagonal matrix. * Any matrix satisfying the condition is said to be

centrosymmetric In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point grou ...

. * 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.

References

{{Matrix classes Matrices (mathematics)

Definition

Properties

Relationships

See also

References