Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular: Other examples include: A totally unimodular matrix[1] (TU matrix) is a matrix for which every square submatrix has determinant 0, +1 or −1.
A totally unimodular matrix need not be square itself.
The converse is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular.
Totally unimodular matrices are extremely important in polyhedral combinatorics and combinatorial optimization since they give a quick way to verify that a linear program is integral (has an integral optimum, when any optimum exists).
Specifically, if A is TU and b is integral, then linear programs of forms like
The unoriented incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching, is totally unimodular (TU).
(The unoriented incidence matrix of a non-bipartite graph is not TU.)
More generally, in the appendix to a paper by Heller and Tompkins,[2] A.J.
be an m by n matrix whose rows can be partitioned into two disjoint sets
Then the following four conditions together are sufficient for A to be totally unimodular: It was realized later that these conditions define an incidence matrix of a balanced signed graph; thus, this example says that the incidence matrix of a signed graph is totally unimodular if the signed graph is balanced.
The converse is valid for signed graphs without half edges (this generalizes the property of the unoriented incidence matrix of a graph).
The constraints of maximum flow and minimum cost flow problems yield a coefficient matrix with these properties (and with empty C).
Thus, such network flow problems with bounded integer capacities have an integral optimal value.
Note that this does not apply to multi-commodity flow problems, in which it is possible to have fractional optimal value even with bounded integer capacities.
The consecutive-ones property: if A is (or can be permuted into) a 0-1 matrix in which for every row, the 1s appear consecutively, then A is TU.
The rows of a network matrix correspond to a tree T = (V, R), each of whose arcs has an arbitrary orientation (it is not necessary that there exist a root vertex r such that the tree is "rooted into r" or "out of r").The columns correspond to another set C of arcs on the same vertex set V. To compute the entry at row R and column C = st, look at the s-to-t path P in T; then the entry is: See more in Schrijver (2003).
Ghouila-Houri showed that a matrix is TU iff for every subset R of rows, there is an assignment
(which is a row vector of the same width as the matrix) has all its entries in
is totally unimodular if and only if every simple arbitrarily-oriented cycle in
consists of alternating forwards and backwards arcs.
Fujishige showed[6] that the matrix is TU iff every 2-by-2 submatrix has determinant in
Seymour (1980)[7] proved a full characterization of all TU matrices, which we describe here only informally.
Seymour's theorem is that a matrix is TU if and only if it is a certain natural combination of some network matrices and some copies of a particular 5-by-5 TU matrix.
Any matrix of the form is not totally unimodular, since it has a square submatrix of determinant −2.
Abstract linear algebra considers matrices with entries from any commutative ring
In this context, a unimodular matrix is one that is invertible over the ring; equivalently, whose determinant is a unit.
Unimodular here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses non-singular to mean matrices that are invertible over the field.