- Integer matrix
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 matrixused in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
: and Are both examples of integer matrices.
Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. The
determinantof 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 theorythat infer properties from determinants thus avoid the traps induced by ill conditioned ("nearly" zero determinant) real or floating pointvalued matrices.
The inverse of an integer matrix is again an integer matrix if and only if the determinant of is exactly or . Integer matrices of determinant form the group , which has far-reaching applications in arithmetic and geometry. For , it is closely related to the
characteristic polynomialof 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.
* [http://mathworld.wolfram.com/IntegerMatrix.html Integer Matrix at MathWorld]
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