In
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 ...
.
[.]
Definition
If is an exchange matrix, then the elements of are
Properties
* Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,
* Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,
* 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:
* 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 ...
:
In particular, is an
involutory matrix; that is,
* The
trace of is 1 if is odd and 0 if is even. In other words:
* 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:
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:
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
) and -1 (with multiplicity
).
* The
adjugate matrix of is:
(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.
See also
*
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
(the first Pauli matrix is a 2 × 2 exchange matrix)
References
{{Matrix classes
Matrices (mathematics)