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 ...

, an integer matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, 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 o ...

, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

Examples

\left(\begin 5 & 2 & 6 & 0\\ 4 & 7 & 3 & 8\\ 5 & 9 & 0 & 4\\ 3 & 1 & 0 & \!\!\!-3\\ 9 & 0 & 2 & 1\end\right)

and

\left(\begin 1 & 5 & 0\\ 0 & 9 & 2\\ 1 & 7 & 3\end\right)

are both examples of integer matrices.

Properties

Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. The determinant of an integer matrix is itself an integer, thus the numerically smallest possible magnitude of the determinant of an invertible integer matrix is one, hence where inverses exist they do not become excessively large (see

condition number In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...

). Theorems from matrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned (''nearly'' zero determinant) real or floating point valued matrices. The inverse of an integer matrix

M

is again an integer matrix if and only if the determinant of

M

equals

1

-1

. Integer matrices of determinant

1

form the group

\mathrm_n(\mathbf)

, which has far-reaching applications in arithmetic and geometry. For

n=2

, it is closely related to the

modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...

. The intersection of the integer matrices with the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

is the group of signed permutation matrices. 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 chara ...

of an integer matrix has integer coefficients. Since the eigenvalues of a matrix are the

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...

of this polynomial, the eigenvalues of an integer matrix are algebraic integers. In dimension less than 5, they can thus be expressed by

radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...

involving integers. Integer matrices are sometimes called ''integral matrices'', although this use is discouraged.

External links

Integer Matrix at MathWorld
{{Matrix classes Matrices

Examples

Properties

See also

External links