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  2. Unimodular matrix - Wikipedia

    en.wikipedia.org/wiki/Unimodular_matrix

    A totally unimodular matrix [1] (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).

  3. Matching polytope - Wikipedia

    en.wikipedia.org/wiki/Matching_polytope

    When G is bipartite, its incidence matrix A G is totally unimodular - every square submatrix of it has determinant 0, +1 or −1. The proof is by induction on k - the size of the submatrix (which we denote by K). The base k = 1 follows from the definition of A G - every element in it is either 0 or 1. For k>1 there are several cases:

  4. Regular matroid - Wikipedia

    en.wikipedia.org/wiki/Regular_matroid

    The regular matroids are the matroids that can be defined from a totally unimodular matrix, a matrix in which every square submatrix has determinant 0, 1, or −1. The vectors realizing the matroid may be taken as the rows of the matrix. For this reason, regular matroids are sometimes also called unimodular matroids. [10]

  5. List of named matrices - Wikipedia

    en.wikipedia.org/wiki/List_of_named_matrices

    Totally unimodular matrix: A matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program. Weighing matrix: A square matrix the entries of which are in {0, 1, −1}, such that AA T = wI for some positive integer w.

  6. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side is integer, has integer basic solutions. This makes the integer program substantially easier to solve.

  7. Totally unimodular matrix - Wikipedia

    en.wikipedia.org/?title=Totally_unimodular...

    This page was last edited on 11 November 2014, at 19:35 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.

  8. Balanced matrix - Wikipedia

    en.wikipedia.org/wiki/Balanced_matrix

    Equivalently, a matrix is totally balanced if and only if it does not contain a submatrix that is the incidence matrix of any cycle (no matter if of odd or even order). This characterization immediately implies that any totally balanced matrix is balanced. [3] Moreover, any 0-1 matrix that is totally unimodular is also balanced. The following ...

  9. Polynomial matrix - Wikipedia

    en.wikipedia.org/wiki/Polynomial_matrix

    A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.