In
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'' (franchi ...
whose entries are all
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 ...
s. Examples include
binary matrices, the
zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed ...
, 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 ...
, and the
adjacency matrices used in
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, 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 a ...
.
Examples
:
and
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
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 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 inpu ...
). Theorems from
matrix theory that infer properties from determinants thus avoid the traps induced by
ill conditioned (''nearly'' zero determinant)
real or
floating point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
valued matrices.
The inverse of an integer matrix
is again an integer matrix if and only if the determinant of
equals
or
. Integer matrices of determinant
form the
group , which has far-reaching applications in arithmetic and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. For
, 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 fraction ...
.
The intersection of the integer matrices with the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
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 ...
of an integer matrix has integer coefficients. Since the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of a matrix are the
roots of this polynomial, the eigenvalues of an integer matrix are
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficien ...
s. 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.
See also
*
GCD matrix
In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix.
Definition
Let S=(x_1, x_2,\ldots, x_n) be a list of positive integers. Then the n\times n matrix (S) having the greatest common divisor \gcd ...
*
Unimodular matrix
*
Wilson matrix
External links
Integer Matrix at MathWorld
{{Matrix classes
Matrices