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 (these are equivalent under Cramer's rule). 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
Z
\operatorname{GL}n(Z)
Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular:
Other examples include:
\det(A ⊗ B)=(\detA)q(\detB)p,
A totally unimodular matrix[1] (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. Equivalently, 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 (TU). 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 (has an integral optimum, when any optimum exists). Specifically, if A is TU and b is integral, then linear programs of forms like
\{minc\topx\midAx\geb,x\ge0\}
\{maxc\topx\midAx\leb\}
\{x\midAx\geb\}
1. 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, A.J. Hoffman and D. Gale prove the following. Let
A
B
C
A
A
A
B
C
A
B
C
2. 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.
3. 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 same holds for columns since the transpose of a TU matrix is also TU.)[3]
4. Every network matrix is TU. The rows of a network matrix correspond to a tree, 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, look at the s-to-t path P in T; then the entry is:
See more in Schrijver (2003).
5. Ghouila-Houri showed that a matrix is TU iff for every subset R of rows, there is an assignment
s:R\to\pm1
\sumrs(r)r
\{0,\pm1\}
6.Hoffman and Kruskalproved the following theorem. Suppose
G
P
G
A
V(G)
P
A
G
7. Suppose a matrix has 0-(
\pm
0,\pm1
8. Seymour (1980) 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.
1. The following matrix is totally unimodular:
A=\left[\begin{array}{rrrrrr} -1&-1&0&0&0&+1\\ +1&0&-1&-1&0&0\\ 0&+1&+1&0&-1&0\\ 0&0&0&+1&+1&-1 \end{array}\right].
This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the maximum flow problem on the following network:
2. Any matrix of the form
A=\left[\begin{array}{ccccc} &\vdots&&\vdots\\ ...b&+1&...b&+1&...b\\ &\vdots&&\vdots\\ ...b&+1&...b&-1&...b\\ &\vdots&&\vdots \end{array}\right].
R
\operatorname{GL}n(R)
k
m
m-k
Rm
Over a field, unimodular has the same meaning as non-singular. 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.