In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a unimodular matrix ''M'' is a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
integer matrix In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integ ...
having
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
+1 or −1. Equivalently, it is an integer matrix that is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
over the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s: there is an integer matrix ''N'' that is its inverse (these are equivalent under
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
). 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
called the ''n'' × ''n''
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
over
, which is denoted
.
Examples of unimodular matrices
Unimodular matrices form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
under
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
, i.e. the following matrices are unimodular:
*
Identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
* The
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
of a unimodular matrix
* The
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of two unimodular matrices
Other examples include:
*
Pascal matrices
*
Permutation matrices
* the three transformation matrices in the ternary
tree of primitive Pythagorean triples
* Certain transformation matrices for
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
,
shearing
Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (a sheep may be said to have been "shorn" o ...
(both with determinant 1) and
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
(determinant −1).
* The unimodular matrix used (possibly implicitly) in
lattice reduction
In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least expone ...
and in the
Hermite normal form In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the H ...
of matrices.
* The
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
of two unimodular matrices is also unimodular. This follows since
where ''p'' and ''q'' are the dimensions of ''A'' and ''B'', respectively.
Total unimodularity
A totally unimodular matrix
(TU matrix) is a matrix for which every square
non-singular submatrix
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
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
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is TU.
Totally unimodular matrices are extremely important in
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.
Research in polyhedral co ...
and
combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combi ...
since they give a quick way to verify that a
linear program
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming i ...
is
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
(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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
polyhedron.
Common totally unimodular matrices
1. The unoriented incidence matrix of a
bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V a ...
, 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
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
and
. Then the following four conditions together are
sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for ''A'' to be totally unimodular:
* Every entry in
is 0, +1, or −1;
* Every column of
contains at most two non-zero (i.e., +1 or −1) 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
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.
A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the no ...
; 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
constraints of
maximum flow
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
The maximum flow problem can be seen as a special case of more complex network flow problems, such ...
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 The multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes.
Definition
Given a flow network \,G(V,E), where edge (u,v) \in E has capacity \,c(u,v). There are \,k comm ...
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 , 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:
* +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 Schrijver (1998).
6.
Hoffman and
Kruskal
proved the following theorem. Suppose
is a
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
without 2-dicycles,
is the set of all
dipath
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes c ...
s 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 −1s are on top, then 0s, then 1s are on the bottom). Fujishige showed
that the matrix is TU iff every 2-by-2 submatrix has determinant in
.
8.
Seymour
Seymour may refer to:
Places Australia
* Seymour, Victoria, a township
* Electoral district of Seymour, a former electoral district in Victoria
* Rural City of Seymour, a former local government area in Victoria
* Seymour, Tasmania, a localit ...
(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.
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
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
The maximum flow problem can be seen as a special case of more complex network flow problems, such ...
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 algebra considers matrices with entries from any
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
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
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
. This
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 (often the integers) which are invertible over that ring, and one uses ''non-singular'' to mean matrices that are invertible over the field.
See also
*
Balanced matrix
*
Regular matroid
In mathematics, a regular matroid is a matroid that can be represented over all fields.
Definition
A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". ...
*
Special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
*
Total dual integrality
*
Hermite normal form In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the H ...
Notes
References
*
*
Alexander Schrijver
Alexander (Lex) Schrijver (born 4 May 1948 in Amsterdam) is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in ...
(1998), ''Theory of Linear and Integer Programming''. John Wiley & Sons, (mathematical)
*
External links
Mathematical Programming Glossary by Harvey J. GreenbergSoftware for testing total unimodularity by M. Walter and K. Truemper
{{Matrix classes
Matrices