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| {{about|matrices whose entries are [[integer number]]s|use of term '''unimodular''' in connection with [[polynomial matrix|polynomial matrices]]|Unimodular polynomial matrix}}
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| 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'' 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 (mathematics)|group]], which is denoted <math>GL_n(\mathbb{Z})</math>.
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| ==Examples of unimodular matrices==
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| Unimodular matrices form a subgroup of the [[general linear group]] under [[matrix multiplication]], i.e. the following matrices are unimodular:
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| * [[Identity matrix]]
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| * The [[Matrix inverse|inverse]] of a unimodular matrix
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| * The [[Matrix multiplication|product]] of two unimodular matrices
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| Further:
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| * The [[Kronecker product]] of two unimodular matrices is also unimodular. This follows since
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| :<math> \det(A \otimes B) = (\det A)^q (\det B)^p, </math>
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| : where ''p'' and ''q'' are the dimensions of ''A'' and ''B'', respectively.
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| Concrete examples include:
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| * [[Symplectic matrix|Symplectic matrices]]
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| * [[Pascal matrix|Pascal matrices]]
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| * [[Permutation matrix|Permutation matrices]]
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| * the three transformation matrices in the ternary [[tree of primitive Pythagorean triples]]
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| ==Total unimodularity==
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| A '''totally unimodular matrix''' <ref>The term was coined by [[Claude Berge]], see {{Citation
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| | last = [[Alan Hoffman (mathematician)|Hoffman]]
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| | first = A.J.
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| | last2 = [[Joseph Kruskal|Kruskal]]
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| | first2 = J.
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| | contribution = Introduction to ''Integral Boundary Points of Convex Polyhedra''
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| | editor-last = M. Jünger et al. (eds.)
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| | title = 50 Years of Integer Programming, 1958-2008
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| | publisher = Springer-Verlag
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| | pages = 49–50
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| | year = 2010}}</ref>
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| (TU matrix) is a matrix for which every square [[invertible matrix|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. The opposite is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular.
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| 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 [[Linear_programming#Integral_linear_programs|integral]] (has an integral optimum, when any optimum exists). Specifically, if ''A'' is TU and ''b'' is integral, then linear programs of forms like <math>\{\min cx \mid Ax \ge b, x \ge 0\}</math> or <math>\{\max cx \mid Ax \le b\}</math> 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. <math>\{x \mid Ax \ge b\}</math>) is integral and thus the feasible region is an [[Linear_programming#Integral_linear_programs|integral]] polyhedron.
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| ===Common totally unimodular matrices===
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| 1. The unoriented incidence matrix of a [[bipartite graph]], which is the coefficient matrix for bipartite [[matching (graph theory)|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,<ref>{{Citation
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| | last = Heller
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| | first = I.
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| | last2 = Tompkins
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| | first2 = C.B.Gh
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| | contribution = An Extension of a Theorem of [[George Dantzig|Dantzig]]'s
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| | editor-last = [[Harold W. Kuhn|Kuhn]]
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| | editor-first = H.W.
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| | editor2-last = [[Albert W. Tucker|Tucker]]
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| | editor2-first = A.W.
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| | title = Linear Inequalities and Related Systems
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| | series = Annals of Mathematics Studies
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| | publisher = Princeton University Press
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| | location = Princeton (NJ)
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| | pages = 247–254
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| | volume = 38
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| | year = 1956 }}</ref>
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| A.J. Hoffman and D. Gale prove the following. Let <math>A</math> be an ''m'' by ''n'' matrix whose rows can be partitioned into two [[disjoint sets]] <math>B</math> and <math>C</math>. Then the following four conditions together are [[Necessary and sufficient conditions|sufficient]] for ''A'' to be totally unimodular:
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| * Every column of <math>A</math> contains at most two non-zero entries;
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| * Every entry in <math>A</math> is 0, +1, or −1;
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| * If two non-zero entries in a column of <math>A</math> have the same sign, then the row of one is in <math>B</math>, and the other in <math>C</math>;
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| * If two non-zero entries in a column of <math>A</math> have opposite signs, then the rows of both are in <math>B</math>, or both in <math>C</math>.
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| It was realized later that these conditions define an incidence matrix of a balanced [[Signed graph#Incidence matrix|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).<ref>T. Zaslavsky (1982), "Signed graphs," ''Discrete Applied Mathematics'' 4, pp. 401–406.</ref> | |
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| 2. The [[Constraint (mathematics)|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.
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| 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.)
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| 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:
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| * +1 if arc ''R'' appears forward in ''P'',
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| * -1 if arc ''R'' appears backwards in ''P'',
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| * 0 if arc ''R'' does not appear in ''P''.
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| See more in Schrijver (2003).
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| 5. Ghouila-Houri showed that a matrix is TU iff for every subset ''R'' of rows, there is an assignment <math>s:R \to \pm1</math> of signs to rows so that the signed sum <math>\sum_{r \in R} s(r)r</math> (which is a row vector of the same width as the matrix) has all its entries in <math>\{0,\pm1\}</math> (i.e. the row-submatrix has [[Discrepancy of hypergraphs|discrepancy]] at most one). This and several other if-and-only-if characterizations are proven in Schrijver (2003).
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| 6.
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| Hoffman and [[Joseph Kruskal|Kruskal]]<ref>{{Citation
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| | last = Hoffman
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| | first = A.J.
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| | last2 = [[Joseph Kruskal|Kruskal]]
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| | first2 = J.B.
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| | contribution = Integral Boundary Points of Convex Polyhedra
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| | editor-last = [[Harold W. Kuhn|Kuhn]]
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| | editor-first = H.W.
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| | editor2-last = [[Albert W. Tucker|Tucker]]
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| | editor2-first = A.W.
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| | title = Linear Inequalities and Related Systems
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| | series = Annals of Mathematics Studies
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| | publisher = Princeton University Press
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| | location = Princeton (NJ)
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| | pages = 223–246
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| | volume = 38
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| | year = 1956 }}</ref>
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| proved the following theorem. Suppose <math>G</math> is a directed graph without any 2-dicycle, <math>P</math> is the set of all dipaths in <math>G</math>, and <math>A</math> is the 0-1 incidence matrix of <math>V(G)</math> versus <math>P</math>. Then <math>A</math> is totally unimodular if and only if every simple arbitrarily-oriented cycle in <math>G</math> consists of alternating forwards and backwards arcs.
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| 7. Suppose a matrix has 0-(<math>\pm</math>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<ref>{{Citation
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| | last = Fujishige
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| | first = Satoru
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| | title = A System of Linear inequalities with a Submodular Function on (0, +/-1 ) Vectors
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| | journal = Linear Algebra and Its Applications
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| | pages = 253–266
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| | volume = 63
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| | year = 1984 }}</ref>
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| that the matrix is TU iff every 2-by-2 submatrix has determinant in <math>0, \pm1</math>.
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| 8. [[Paul Seymour (mathematician)|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.
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| ===Concrete examples===
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| 1. The following matrix is totally unimodular:
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| :<math>A=\begin{bmatrix}
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| -1 & -1 & 0 & 0 & 0 & +1\\
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| +1 & 0 & -1 & -1 & 0 & 0\\
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| 0 & +1 & +1 & 0 & -1 & 0\\
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| 0 & 0 & 0 & +1 & +1 & -1\\
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| \end{bmatrix}.</math>
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| This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the [[Max-flow min-cut theorem|maximum flow]] problem on the following network:
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| [[Image:Graph for example adjacency matrix.svg]]
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| 2. Any matrix of the form
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| :<math>A=\begin{bmatrix}
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| \vdots & \vdots & \vdots & \vdots & \vdots \\
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| \dotsb & +1 & \dotsb & +1 & \dotsb\\
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| \vdots & \vdots & \vdots & \vdots & \vdots \\
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| \dotsb & +1 & \dotsb & -1 & \dotsb\\
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| \vdots & \vdots & \vdots & \vdots & \vdots \\
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| \end{bmatrix}.</math>
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| is ''not'' totally unimodular, since it has a square submatrix of determinant -2.
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| == Abstract linear algebra ==
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| [[abstract algebra|Abstract linear algebra]] considers matrices with entries from any [[commutative]] [[ring (mathematics)|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 (ring theory)|unit]]. This [[group (mathematics)|group]] is denoted <math>GL_n R\,</math>.
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| Over a [[field (mathematics)|field]], ''unimodular'' has the same meaning as ''[[invertible matrix|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.
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| ==See also==
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| *[[Balanced matrix]]
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| *[[Regular matroid]]
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| *[[Special linear group]]
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| *[[Total dual integrality]]
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| == Notes ==
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| <references />
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| == References ==
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| *{{Citation
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| | last = Papadimitriou
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| | first = Christos H.
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| | last2 = Steiglitz
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| | first2 = Kenneth
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| | year = 1998
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| | title = Combinatorial Optimization: Algorithms and Complexity
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| | publisher = Dover Publications (Section 13.2)
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| | location = Mineola, N.Y.
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| | note = The 'section 13.2' is on the wrong place, but I don't know how to fix it}}
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| * [[Alexander Schrijver]] (1998), ''Theory of Linear and Integer Programming''. John Wiley & Sons, ISBN 0-471-98232-6 (mathematical)
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| * {{Citation|author = Alexander Schrijver|authorlink=Alexander Schrijver | year = 2003 | title = Combinatorial Optimization: Polyhedra and Efficiency | publisher = Springer}}
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| ==External links==
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| * [http://glossary.computing.society.informs.org/index.php?page=U.html Mathematical Programming Glossary by Harvey J. Greenberg]
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| * [http://mathworld.wolfram.com/UnimodularMatrix.html Unimodular Matrix from MathWorld]
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| * [http://www.math.uni-magdeburg.de/~walter/TUtest/ Software for testing total unimodularity by M. Walter and K. Truemper]
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| [[Category:Matrices]]
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