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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a unimodular matrix ''M'' is a
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
integer matrix having
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
+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 ...
over the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 of ...
). 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 iden ...
called the ''n'' × ''n''
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
over \mathbb, which is denoted \operatorname_n(\mathbb).


Examples of unimodular matrices

Unimodular matrices form a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
under
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
, 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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
* The inverse of a unimodular matrix * The product of two unimodular matrices Other examples include: * Pascal matrices *
Permutation matrices In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...
* the three transformation matrices in the ternary tree of primitive Pythagorean triples * Certain transformation matrices for
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
,
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 (depending upon dialect, a sheep may be sai ...
(both with determinant 1) and reflection (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 expon ...
and in the Hermite normal form 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 specialization of the tensor product (which is denoted by the same symbol) from vector ...
of two unimodular matrices is also unimodular. This follows since \det(A \otimes B) = (\det A)^q (\det B)^p, 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 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 In linear algebra, the transpose of a Matrix (mathematics), 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 ...
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 combina ...
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 and objective are represented by linear relationships. Linear ...
is
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
(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 is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
polyhedron.


Common totally unimodular matrices

1. The unoriented incidence matrix of a
bipartite graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...
, 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 be an ''m'' by ''n'' matrix whose rows can be partitioned into two
disjoint sets In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
B and C. Then the following four conditions together are sufficient for ''A'' to be totally unimodular: * Every entry in A is 0, +1, or −1; * Every column of A contains at most two non-zero (i.e., +1 or −1) entries; * If two non-zero entries in a column of A have the same sign, then the row of one is in B, and the other in C; * If two non-zero entries in a column of A have opposite signs, then the rows of both are in B, or both in C. 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 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.) 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 s:R \to \pm1 of signs to rows so that the signed sum \sum_ s(r)r (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 G is a directed graph without 2-dicycles, P is the set of all dipaths in G, and A is the 0-1 incidence matrix of V(G) versus P. Then A is totally unimodular if and only if every simple arbitrarily-oriented cycle in G consists of alternating forwards and backwards arcs. 7. Suppose a matrix has 0-(\pm1) 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 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.


Concrete examples

1. The following matrix is totally unimodular: :A=\left begin -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\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 & \vdots & & \vdots \\ \dotsb & +1 & \dotsb & +1 & \dotsb \\ & \vdots & & \vdots \\ \dotsb & +1 & \dotsb & -1 & \dotsb \\ & \vdots & & \vdots \end\right 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. Perhaps most familiar as a pr ...
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
R, 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 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 iden ...
is denoted \operatorname_n(R). A rectangular k-by-m matrix is said to be unimodular if it can be extended with m-k rows in R^m to a unimodular square matrix. Over a field, ''unimodular'' has the same meaning as ''
non-singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' 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 *
Special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...
* Total dual integrality * Hermite normal form


Notes


References

* * Alexander Schrijver (1998), ''Theory of Linear and Integer Programming''. John Wiley & Sons, (mathematical) *


External links


Mathematical Programming Glossary by Harvey J. Greenberg



Software for testing total unimodularity by M. Walter and K. Truemper
{{Matrix classes Matrices (mathematics)