Unimodularity

84. Unimodularity

84.1. Unimodular Matrix

Integer square matrix whose determinant is +1 or −1:

det (𝐴) = \pm 1

Always invertible, and its inverse is also an integer matrix
Applying such a matrix (transformation) to an integer vector gives another integer vector

84.2. Totally Unimodular Matrix

A totally unimodular (TU) matrix is a (not necessarily square) matrix in which every square submatrix (determinant of any square submatrix of any size) has determinant in:

{- 1, 0, + 1}

𝐴 = [\begin{matrix}  \end{matrix}]

det ([\begin{matrix}  \end{matrix}]) = 0 det ([\begin{matrix}  \end{matrix}]) = 0