In
mathematics, an integer matrix is a
matrix 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 languag ...
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
In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below:
:J_2 = \begin
1 & 1 \\
1 & 1
\end;\quad
J_3 = \begin
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end;\quad
...
, the
identity matrix, 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.
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 a ...
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). Theorems from
matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
that infer properties from determinants thus avoid the traps induced by
ill conditioned (''nearly'' zero determinant)
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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 b ...
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, 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 ...
. For
, it is closely related to the
modular group.
The intersection of the integer matrices with the
orthogonal group is the group of
signed permutation matrices
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the non ...
.
The
characteristic polynomial 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 denoted ...
s 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 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 coefficients ...
s. In dimension
less than 5, they can thus be expressed by
radicals 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
In ...
*
Unimodular matrix
In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equiv ...
*
Wilson matrix
External links
Integer Matrix at MathWorld
{{Matrix classes
Matrices