mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 matrice ...

, 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 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. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...

.. :

J_=\begin
0 & 1 \\
1 & 0
\end;\quad J_ = \begin
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end;
\quad J_ = \begin
  0      & 0      & \cdots & 0      & 0      & 1      \\
  0      & 0      & \cdots & 0      & 1      & 0      \\
  0      & 0      & \cdots & 1      & 0      & 0      \\
  \vdots & \vdots &        & \vdots & \vdots & \vdots \\ 
  0      & 1      & \cdots & 0      & 0      & 0      \\
  1      & 0      & \cdots & 0      & 0      & 0      
\end.

Definition

If ''J'' is an ''n'' × ''n'' exchange matrix, then the elements of ''J'' are

J_ = \begin 
1, & i + j = n + 1 \\
0, & i + j \ne n + 1\\
\end

Properties

* Exchange matrices are

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

; that is, ''J''_''n''^T = ''J''_''n''. * For any

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

''k'', ''J''_''n''^''k'' = ''I'' if ''k'' is

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

and ''J''_''n''^k = ''J''_''n'' if ''k'' is

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

. 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, the trace of ''J''_''n'' equals

n\bmod 2

. * The

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

of ''J''_''n'' equals

(-1)^

. 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 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 c ...

of ''J''_''n'' is

\det(\lambda I- J_n) = \big((\lambda+1)(\lambda-1)\big)^

when ''n'' is even, and

(\lambda-1)^(\lambda+1)^

when ''n'' is odd. * The

adjugate matrix In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...

of ''J''_''n'' is

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

Relationships

* An exchange matrix is the simplest

anti-diagonal matrix In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal (sometimes Harrison diagonal, secon ...

. * Any matrix ''A'' satisfying the condition ''AJ = JA'' 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 gr ...

. * 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.

References

Matrices {{Linear-algebra-stub

Definition

Properties

Relationships

See also

References