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2013-11-14T14:30:11Z

84.44.7.210: Undid revision 581631861 by 84.44.7.210 (talk)

In [[mathematics]], an '''integer matrix''' is a [[matrix (mathematics)|matrix]] whose entries are all [[integer]]s. Examples include [[binary matrix|binary matrices]], the [[zero matrix]], the [[unit matrix]], and the [[adjacency matrix|adjacency matrices]] used in [[graph theory]], amongst many others. Integer matrices find frequent application in [[combinatorics]].

==Examples==
:<math>\left(\begin{array}{cccc} 5 & 2 & 6 & 0\\ 4 & 7 & 3 & 8\\ 5 & 9 & 0 & 4\\ 3 & 1 & 0 & -3\\ 9 & 0 & 2 & 1\end{array}\right)</math>    and     <math>\left(\begin{array}{ccc} 1 & 5 & 0\\ 0 & 9 & 2\\ 1 & 7 & 3\end{array}\right)</math>
are both examples of integer matrices.

==Properties==
[[matrix inverse|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]]). Theorems from [[Matrix (mathematics)|matrix theory]] that infer properties from determinants thus avoid the traps induced by [[ill conditioned matrix|ill conditioned]] (''nearly'' zero determinant) [[real numbers|real]] or [[floating point]] valued matrices.

The inverse of an integer matrix <math>M</math> is again an integer matrix if and only if the determinant of <math>M</math> is exactly <math>1</math> or <math>-1</math>. Integer matrices of determinant <math>\pm 1</math> form the group <math>\mathrm{GL}_n(\mathbf{Z})</math>, which has far-reaching applications in arithmetic and geometry. For <math>n=2</math>, 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]].

The [[characteristic polynomial]] of an integer matrix has integer coefficients. Since the [[eigenvalue]]s of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are [[algebraic integers]]. In dimension [[Abel-Ruffini theorem|less than 5]], they can thus be expressed by [[Nth root|radicals]] involving integers.

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

==See also==
*[[Unimodular matrix]]

==External links==

*[http://mathworld.wolfram.com/IntegerMatrix.html Integer Matrix at MathWorld]

[[Category:Matrices]]

{{Linear-algebra-stub}}

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