Integer matrix

Integer matrix

In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include the binary matrix; the zero matrix; the unit matrix; the adjacency matrix used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.

For example:

:left(egin{array}{cccc} 5 & 2 & 6 & 0\ 4 & 7 & 3 & 8\ 5 & 9 & 0 & 4\ 3 & 1 & 0 & 3\ 9 & 0 & 2 & 1end{array} ight) and left(egin{array}{ccc} 1 & 5 & 0\ 0 & 9 & 2\ 1 & 7 & 3end{array} ight)Are both examples of integer matrices.


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 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 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 is exactly 1 or -1. Integer matrices of determinant pm 1 form the group mathrm{GL}(n,mathbf{Z}), which has far-reaching applications in arithmetic and geometry. For n=2, it is closely related to the modular group.

The characteristic polynomial of an integer matrix has integer coefficients. Since the eigenvalues of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are algebraic integers. 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]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Matrix - получить на Академике рабочий купон на скидку Строительный Двор или выгодно matrix купить с бесплатной доставкой на распродаже в Строительный Двор

  • Matrix Template Library — Operating system Linux, Unix, Mac OS X, Windows Available in C++ Type Scientific software library Lice …   Wikipedia

  • Matrix mechanics — Quantum mechanics Uncertainty principle …   Wikipedia

  • Integer factorization — In number theory, integer factorization is the way of breaking down a composite number into smaller non trivial divisors, which when multiplied together equal the original integer.When the numbers are very large, no efficient integer… …   Wikipedia

  • Matrix Math Extensions — Intel Prozessor mit MMX Die Multi Media Extension (kurz MMX) ist eine Anfang 1997 von Intel auf den Markt gebrachte Rechnerarchitektur, die es erlaubt, größere Datenmengen parallelisiert und somit schneller zu verarbeiten. Die… …   Deutsch Wikipedia

  • Matrix multiplication — In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n by m matrix and B is an m by p matrix, the result AB of their multiplication is an n by p matrix defined only if… …   Wikipedia

  • Matrix exponential — In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.… …   Wikipedia

  • Matrix representation — This article is about the layout of matrices in the memory of computers. For the representation of groups and algebras by matrices in linear algebra, see representation theory. Matrix representation is a method used by a computer language to… …   Wikipedia

  • Integer lattice — In mathematics, the n dimensional integer lattice (or cubic lattice), denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are n tuples of integers. The two dimensional integer lattice is also called the square lattice, or… …   Wikipedia

  • Matrix pencil — If are complex matrices for some nonnegative integer l, and (the zero matrix), then the matrix pencil of degree l is the matrix valued function defined on the complex numbers A particular case is a linear matrix pencil …   Wikipedia

  • Matrix grammar — A matrix grammar is a formal grammar in which instead of single productions, productions are grouped together into finite sequences. A production cannot be applied separately, it must be applied in sequence. In the application of such a sequence… …   Wikipedia