unimodularity

Unimodularity is a term used in linear algebra and lattice theory to describe certain integer matrices. A square integer matrix is called unimodular if its determinant is plus or minus one. Such matrices are invertible over the integers, and their inverses are themselves integer matrices. Consequently, unimodular matrices belong to the group GL(n, Z), the automorphisms of the integer lattice Z^n.

Key properties include that unimodular matrices map the integer lattice to itself in a bijective, structure-preserving way. They preserve lattice volume up to sign: a determinant of +1 preserves orientation, while a determinant of -1 reverses it. The inverse of a unimodular matrix is also unimodular, ensuring that lattice relations remain integral after transformation. In practical terms, unimodular matrices can be generated by elementary row operations that add integer multiples of one row to another, swap rows, or multiply a row by -1.

Applications arise in solving systems of linear Diophantine equations and in integer programming. When constraint matrices are unimodular and right-hand sides are integral, certain linear programs yield integral optimal solutions without explicit rounding. A related stronger concept is total unimodularity, where every square submatrix has determinant in {0, ±1}; total unimodularity guarantees integrality of all basic feasible solutions in linear programs with integral data.

Related ideas include unimodular lattices, which are lattices with particularly symmetric inner products, and the broader notion of automorphisms of the integer lattice. Common examples of unimodular matrices are identity matrices and permutation matrices, as well as matrices obtained by multiplying rows or columns by −1.