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 ...

, an integer 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 ...

whose entries are all

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 ...

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 followe ...

, 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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

, and the adjacency matrices used in

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, 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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

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 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 an integer matrix

M

is itself an integer, and the adjugate matrix of an integer matrix is also integer matrix. Thus, the determinant of an invertible integer matrix is

1

-1

, and hence where inverses exist they do not become excessively large (see condition number). Theorems from

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. ...

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 Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

. For

n=2

, it is closely related to the modular group. 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 ...

of an integer matrix has integer coefficients. Since the

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 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 focusin ...

of this polynomial, the eigenvalues of an integer matrix are

algebraic integer In algebraic number theory, an algebraic integer is a complex number that 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.

External links

Integer Matrix at MathWorld

Reference

Matrices (mathematics) {{Matrix classes Matrices (mathematics)

Examples

Properties

See also

External links

Reference