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 in

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 ...

and

matrix theory In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. ...

, a centrosymmetric matrix is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

which is symmetric about its center.

Formal definition

An matrix is centrosymmetric when its entries satisfy

A_ = A_ \quad \texti,j \in \.

Alternatively, if denotes the

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 an ...

with 1 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 0 elsewhere:

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

then a matrix is centrosymmetric

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

Examples

* All 2 × 2 centrosymmetric matrices have the form

\begin a & b \\ b & a \end.

* All 3 × 3 centrosymmetric matrices have the form

\begin a & b & c \\ d & e & d \\ c & b & a \end.

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 ...

Toeplitz matrices are centrosymmetric.

Algebraic structure and properties

*If and are centrosymmetric matrices over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, then so are and for any in . Moreover, the

matrix product In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

is centrosymmetric, since . Since 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 ...

is also centrosymmetric, it follows that the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of centrosymmetric matrices over forms a

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

of the

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

of all matrices. *If is a centrosymmetric matrix with an -dimensional

eigenbasis 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 c ...

, then its

eigenvector 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 ...

s can each be chosen so that they satisfy either or where is the exchange matrix. *If is a centrosymmetric matrix with distinct

eigenvalue 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 ...

s, then the matrices that commute with must be centrosymmetric. *The maximum number of unique elements in an centrosymmetric matrix is ::

\frac.

Related structures

An matrix is said to be ''skew-centrosymmetric'' if its entries satisfy

A_ = -A_ \quad \texti,j \in \.

Equivalently, is skew-centrosymmetric if , where is the exchange matrix defined previously. The centrosymmetric relation lends itself to a natural generalization, where is replaced with an

involutory matrix In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix \bold A_ is an involution if and only if \bold A^2 = \bold I, where \bold I is the n \times n identity matrix. Involutory matri ...

(i.e., ) or, more generally, a matrix satisfying for an

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 ...

. The inverse problem for the commutation relation of identifying all involutory that commute with a fixed matrix has also been studied.

centrosymmetric matrices are sometimes called bisymmetric matrices. When the

ground field In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion. Use It is used in various areas of algebra: In linear algebra In linear algebra, the concept of a vector space may be developed over any field. In algebr ...

is the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. A similar result holds for

Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...

centrosymmetric and skew-centrosymmetric matrices.

References

External links

Centrosymmetric matrix
on

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