In 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 matric ...

, 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 theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...

, 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 ...

.. :

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 definit ...

; 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 language ...

''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 ga ...

and ''J''_''n''^k = ''J''_''n'' if ''k'' is odd. In particular, ''J''_''n'' is an

involutory matrix In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the ''n'' × ''n'' identity matrix. Involutory matr ...

; that is, ''J''_''n''⁻¹ = ''J''_''n''. * The

trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''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 ...

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 ...

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 differ ...

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

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

Relationships

* An exchange matrix is the simplest anti-diagonal matrix. * 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 grou ...

. * 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