Integer matrix

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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.

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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 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 will be integers or surds involving integers, when the dimension of the matrix is strictly less than 5. Quintic and higher polynomials have no closed form roots in general, so no such assertion can be made.

Note that the inverse of an integer matrix is not generally an integer matrix.

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