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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 that is its inverse (these are equivalent under Cramer's rule ).
In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinant det(A) is a nonzero constant [1].
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1. A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
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]
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:
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.
Unimodular polynomial matrix; Unimodular form; Unimodular group This page was last edited on 30 December 2019, at 17:59 (UTC). Text is available under the Creative ...
A totally unimodular matrix is a matrix for which every square submatrix is non-singular and unimodular? Ngvrnd 15:44, 9 May 2011 (UTC) Indeed such a matrix is totally unimodular. But if you request that every submatrix is singular, then also the 1x1 matrices are, that is, the matrix is a zero matrix. But it is totally unimodular. Your 2nd ...