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 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 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 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 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.
Read more about this topic: Integer Matrix
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)