Unimodular matrix
Encyclopedia
In mathematics
, a unimodular matrix M is a square integer matrix
with determinant
+1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse (these are equivalent under Cramer's rule
). Thus every equation Mx = b, where M and b are both integer, and M is unimodular, has an integer solution. The unimodular matrices of order n form a group
, which is denoted
.
under matrix multiplication
, i.e. the following matrices are unimodular:
Further:
Concrete examples include:
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
or 
have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. 
) is integral and thus the feasible region is an integral polyhedron.
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:
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).
2. The constraint
s 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 problem
s, 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.)
4. Every network matrix 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).
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 
(which is a row vector of the same width as the matrix) has all its entries in 
(i.e. the row-submatrix has discrepancy at most one). This and several other if-and-only-if characterizations are proven in Schrjver (2003).
6.
Hoffman and Kruskal
proved the following theorem. Suppose
is a directed graph without any 2-dicycle, 
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-(
1) entries and in each column, the entries are non-decreasing from top to bottom (so all -1's are on top, then 0's, then 1's are on the bottom). Fujishige showed
that the matrix is TU iff every 2-by-2 submatrix has determinant in
.
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 10-by-10 TU matrix.
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
is not totally unimodular, since it has a square submatrix of determinant -2.
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
.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a unimodular matrix M is a square integer matrix
Integer matrix
In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the unit matrix, and the adjacency matrices used in graph theory, amongst many others...
with determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
+1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse (these are equivalent under Cramer's rule
Cramer's rule
In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
). Thus every equation Mx = b, where M and b are both integer, and M is unimodular, has an integer solution. The unimodular matrices of order n form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, which is denoted
.Examples of unimodular matrices
Unimodular matrices form a subgroup of the general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
under matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
, i.e. the following matrices are unimodular:
- Identity matrixIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
- The inverse of a unimodular matrix
- The productMatrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
of two unimodular matrices
Further:
- The Kronecker productKronecker productIn mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...
of two unimodular matrices is also unimodular. This follows since
- where p and q are the dimensions of A and B, respectively.
Concrete examples include:
- Symplectic matrices
- Pascal matricesPascal matrixIn mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, a lower-triangular matrix, or a symmetric matrix...
- Permutation matricesPermutation matrixIn mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
Total unimodularity
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A totally unimodular matrix need not be square itself. From the definition it follows that any totally unimodular matrix has only 0, +1 or −1 entries.Totally unimodular matrices are extremely important in polyhedral combinatorics
Polyhedral combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher dimensional convex polytopes....
and combinatorial optimization
Combinatorial optimization
In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible...
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
or have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. ) 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 (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
be an m by n matrix whose rows can be partitioned into two disjoint sets and . Then the following four conditions together are sufficientNecessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...
for A to be totally unimodular:
- Every column of
contains at most two non-zero entries; - Every entry in
is 0, +1, or −1; - 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 (this generalizes the property of the unoriented incidence matrix of a graph).
2. The constraint
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
s 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 problem
Multi-commodity flow problem
The multi-commodity flow problem is a network flow problem with multiple commodities flowing through the network, with different source and sink nodes.-Definition:Given a flow network \,G, where edge \in E has capacity \,c...
s, 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.)
4. Every network matrix 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:
- +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 (2003).
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 (which is a row vector of the same width as the matrix) has all its entries in (i.e. the row-submatrix has discrepancy at most one). This and several other if-and-only-if characterizations are proven in Schrjver (2003).6.
Hoffman and Kruskal
Joseph Kruskal
Joseph Bernard Kruskal, Jr. was an American mathematician, statistician, computer scientist and psychometrician. He was a student at the University of Chicago and at Princeton University, where he completed his Ph.D. in 1954, nominally under Albert W...
proved the following theorem. Suppose
is a directed graph without any 2-dicycle, 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-(
1) entries and in each column, the entries are non-decreasing from top to bottom (so all -1's are on top, then 0's, then 1's are on the bottom). Fujishige showedthat the matrix is TU iff every 2-by-2 submatrix has determinant in
.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 10-by-10 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
Max-flow min-cut theorem
In optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the minimum capacity which when removed in a specific way from the network causes the situation that no flow can pass from the source to the...
problem on the following network:
2. Any matrix of the form
is not totally unimodular, since it has a square submatrix of determinant -2.
Abstract linear algebra
Abstract linear algebraAbstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
considers matrices with entries from any commutative ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, 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
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
. This group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is denoted
.Over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, 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.