Integer matrix

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

## Contents

* 1 Examples
* 2 Properties
* 3 See also
* 4 External links

## Examples

(526047385904310−39021) and  (150092173)

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 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 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 or −1. Integer matrices of determinant 1 form the group SLn(𝐙), which has far-reaching applications in arithmetic and geometry. For n=2, 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 eigenvalues of a matrix are the roots of this 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.

## See also

* GCD matrix
* Unimodular matrix
* Wilson matrix

## External links

* Integer Matrix at MathWorld

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Matrix classes

Explicitly constrained entries|

* (0,1)
* Alternant
* Anti-diagonal
* Anti-Hermitian
* Anti-symmetric
* Arrowhead
* Band
* Bidiagonal
* Binary
* Bisymmetric
* Block-diagonal
* Block
* Block tridiagonal
* Boolean
* Cauchy
* Centrosymmetric
* Conference
* Complex Hadamard
* Copositive
* Diagonally dominant
* Diagonal
* Discrete Fourier Transform
* Elementary
* Equivalent
* Frobenius
* Generalized permutation
* Hadamard
* Hankel
* Hermitian
* Hessenberg
* Hollow
* Integer
* Logical
* Markov
* Metzler
* Monomial
* Moore
* Nonnegative
* Partitioned
* Parisi
* Pentadiagonal
* Permutation
* Persymmetric
* Polynomial
* Positive
* Quaternionic
* Sign
* Signature
* Skew-Hermitian
* Skew-symmetric
* Skyline
* Sparse
* Sylvester
* Symmetric
* Toeplitz
* Triangular
* Tridiagonal
* Unitary
* Vandermonde
* Walsh
* Z

Constant|

* Exchange
* Hilbert
* Identity
* Lehmer
* Of ones
* Pascal
* Pauli
* Redheffer
* Shift
* Zero

Conditions on eigenvalues or eigenvectors|

* Companion
* Convergent
* Defective
* Diagonalizable
* Hurwitz
* Positive-definite
* Stability
* Stieltjes

Satisfying conditions on products or inverses|

* Congruent
* Idempotent or Projection
* Invertible
* Involutory
* Nilpotent
* Normal
* Orthogonal
* Orthonormal
* Singular
* Unimodular
* Unipotent
* Totally unimodular
* Weighing

With specific applications|

* Adjugate
* Alternating sign
* Augmented
* Bézout
* Carleman
* Cartan
* Circulant
* Cofactor
* Commutation
* Confusion
* Coxeter
* Derogatory
* Distance
* Duplication
* Elimination
* Euclidean distance
* Fundamental (linear differential equation)
* Generator
* Gramian
* Hessian
* Householder
* Jacobian
* Moment
* Payoff
* Pick
* Random
* Rotation
* Seifert
* Shear
* Similarity
* Symplectic
* Totally positive
* Transformation
* Wedderburn
* X–Y–Z

Used in statistics|

* Bernoulli
* Centering
* Correlation
* Covariance
* Design
* Dispersion
* Doubly stochastic
* Fisher information
* Hat
* Precision
* Stochastic
* Transition

Used in graph theory|

* Adjacency
* Biadjacency
* Degree
* Edmonds
* Incidence
* Laplacian
* Seidel adjacency
* Skew-adjacency
* Tutte

Used in science and engineering|

* Cabibbo–Kobayashi–Maskawa
* Density
* Fundamental (computer vision)
* Fuzzy associative
* Gamma
* Gell-Mann
* Hamiltonian
* Irregular
* Overlap
* S
* State transition
* Substitution
* Z (chemistry)

Related terms|

* Jordan canonical form
* Linear independence
* Matrix exponential
* Matrix representation of conic sections
* Perfect matrix
* Pseudoinverse
* Quaternionic matrix
* Row echelon form
* Wronskian

* List of matrices
* Category:Matrices

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