Unimodular matrix


In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse. Thus every equation, where M and b both have integer components and M is unimodular, has an integer solution. The n × n unimodular matrices form a group called the n × n general linear group over, which is denoted.

Examples of unimodular matrices

Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular:
Other examples include:

Total unimodularity

A totally unimodular matrix
is a matrix for which every square submatrix has determinant 0, +1 or −1. A totally unimodular matrix need not be square itself. From the definition it follows that any submatrix of a totally unimodular matrix is itself totally unimodular. Furthermore it follows that any TU matrix has only 0, +1 or −1 entries. The converse is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular. A matrix is TU if and only if its transpose is TU.
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. Specifically, if A is TU and b is integral, then linear programs of forms like or have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region is integral and thus the feasible region is an integral polyhedron.

Common totally unimodular matrices

1. The unoriented incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching, is totally unimodular. More generally, in the appendix to a paper by Heller and Tompkins, A.J. Hoffman and D. Gale prove the following. Let be an m by n matrix whose rows can be partitioned into two disjoint sets and . Then the following four conditions together are sufficient for A to be totally unimodular:
  • Every entry in is 0, +1, or −1;
  • Every column of contains at most two non-zero entries;
  • If two non-zero entries in a column of have the same sign, then the row of one is in, and the other in ;
  • If two non-zero entries in a column of have opposite signs, then the rows of both are in, or both in.
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.
2. The constraints of maximum flow and minimum cost flow problems yield a coefficient matrix with these properties. 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.
3. The consecutive-ones property: if A is a 0-1 matrix in which for every row, the 1s appear consecutively, then A is TU.
4. Every network matrix is TU. The rows of a network matrix correspond to a tree, each of whose arcs has an arbitrary orientation.The columns correspond to another set C of arcs on the same vertex set V. To compute the entry at row R and column, look at the s-to-t path P in T; then the entry is:
  • +1 if arc R appears forward in P,
  • −1 if arc R appears backwards in P,
  • 0 if arc R does not appear in P.
See more in Schrijver.
5. Ghouila-Houri showed that a matrix is TU iff for every subset R of rows, there is an assignment of signs to rows so that the signed sum has all its entries in . This and several other if-and-only-if characterizations are proven in Schrijver.
6.
Hoffman and Kruskal
proved the following theorem. Suppose is a directed graph without 2-dicycles, is the set of all dipaths in, and is the 0-1 incidence matrix of versus. Then is totally unimodular if and only if every simple arbitrarily-oriented cycle in consists of alternating forwards and backwards arcs.
7. Suppose a matrix has 0- entries and in each column, the entries are non-decreasing from top to bottom. Fujishige showed
that the matrix is TU iff every 2-by-2 submatrix has determinant in.
8. Seymour 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.

Concrete examples

1. The following matrix is totally unimodular:
This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the maximum flow problem on the following network:
Image:Graph for example adjacency matrix.svg
2. Any matrix of the form
is not totally unimodular, since it has a square submatrix of determinant −2.

Abstract linear algebra

Abstract linear algebra considers matrices with entries from any commutative ring, not limited to the integers. In this context, a unimodular matrix is one that is invertible over the ring; equivalently, whose determinant is a unit. This group is denoted. A rectangular -by- matrix is said to be unimodular if it can be extended with rows in to a unimodular square matrix.
Over a field, unimodular has the same meaning as non-singular. Unimodular here refers to matrices with coefficients in some ring which are invertible over that ring, and one uses non-singular to mean matrices that are invertible over the field.