Questions tagged [unimodular-matrices]

A unimodular matrix is a matrix whose determinant is $\pm 1$. Often, but not always, its entries are integers.

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Why is $PSL(2,F)$ such an important group?

I am currently writing a final project on simple groups and my professor said I should focus on "the most important simple group", $PSL(2,F)$. I have also read in Aluffi's algebra book that $...
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Let $A \in \mathbb{Q}^{m \times n}$ with full row rank. Then the matrix $B$ in $A$’s Hermite Normal Form $[B \quad 0]$ is invertible.

My lecture introduces several new definitions and theorems as below. The part about the matrix $B$ is a passing remark that I’d like to have some more elaboration. Definition 1. Let $A$ be a $m \...
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Proof of Hoffman & Kruskal's theorem on Unimodularity and Integrality.

I'm reading the following proof. Thm. (Unimodularity and Integrality, Hoffman & Kruskal [1956]): Let $A \in \mathbb{R}^{m \times n}, \operatorname{rank} A=n .$ The following are equivalent: a) $A$...
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Can every symmetric, unimodular and positive definite $G\in\mathbb{Z}^{n\times n}$ be written as $G=U^TU$?

Let $G\in\mathbb{Z}^{n\times n}$ be symmetric, unimodular and positive definite. Does there exist a unimodular matrix $U\in\mathbb{Z}^{n\times n}$ such that $G=U^TU$? I now that the result is true if $...
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Parametrization of all unimodular matrices of rank $n$?

For a given rank $n$, are there parametrization families that cover all possible unimodular matrices of rank $n$? For example for $n=3$ Wolfram gives an example of one parametrization family.
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71 views

Appending TU matrix with the identity or zero matrix still give us another TU matrix.

If $A$ was a totally unimodular matrix, then show the matrix obtained by augmenting with the identity matrix $[A|I]$ is TU. Also show the matrix obtained by augmenting with the zero matrix matrix [$A|...
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88 views

Prove that $3 \times 3$ matrices under certain conditions form a group

Let $S = \{M \in M_3(\mathbb{Z})\mid M^T \Omega M = \Omega \}$, where $$\Omega = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$ I'm trying to prove ...
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31 views

Total Unimodularity of set of equality and inequality constraints by partitioning of rows

Consider binary decision variables $x_{ij}$ and $y_j$ where $ i \in \{1,2,\ldots,I\}$ and $ j \in \{1,2,\ldots,J\}$ for fixed integers $I$ and $J$. Consider the following feasibility prolem: \begin{...
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Is there a standard name for the groups $SL(n,\mathbb Z)$ or $GL(n,\mathbb Z)$?

Is there a standard name for the set of all $n\times n$ integer matrices with determinant $1$ (special linear) or $\pm1$ (general linear)? The matrices themselves are called "unimodular", but ...
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How can you verify that a 3 by 3 unimodular matrix generates an infinite number of Fermat near misses?

I am interested in finding 3 by 3 Ramanujan-Hirschhorn matrices. By definition, a Ramanujan-Hirschhorn matrix is a 3 by 3 matrix which produces an infinite number of Fermat near misses. Ramanujan in ...
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Making Matrix Totally Unimodular

Is there a way I can rewrite the following matrices to make the matrix (A) to be totally unimodular and still maintain the relevance of the equations. Thanks.
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How to prove that the matrix is totally unimodular?

I have the following matrix $$A=\pmatrix{1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\0&0&0&0&...
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How can I compute a 3 by 3 unimodular matrix which produces an infinite number of Fermat near misses?

I am interested in finding 3 by 3 Ramanujan-Hirschhorn matrices. By definition, a Ramanujan-Hirschhorn matrix is a 3 by 3 matrix which produces an infinite number of Fermat near misses. Ramanujan ...
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Totally Unimodular Matrices

Given a non-squared $M$, and a square $N$ totally unimodular matrices (TUM), is it true that if I consider $$ HM = MC $$ then is it so that $H$ must be TUM? I know that $MC$ must be TUM as the ...
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Reference for a result in Abelian Group Theory

I'm looking for a reference of the following result: Suppose $G$ is a non trivial subgroup of $\mathbb{Z}^n$ of the form $H \cap \mathbb{Z}^n$ where $H$ is a hyperplane of $\mathbb{R}^n$. Let $rank(...
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Ring of invariants of unimodular matrices acting on real square matrices

Suppose that $M(\Bbb R,n)$ is the set of all $n \times n$ square real matrices. The special linear group $\text{SL}(\Bbb R,n)$ acts on this group via right-multiplication. It is easy to see that the ...
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Proof $D_2^{-1}D_1$ is a unimodular matrix.

Consider $G=N_1D_1^{-1}=N_2D_2^{-1}$ and the both descriptions of are composed of coprime matrix, prove that $D_2^{-1}D_1$ is a unimodular matrix. The matrices $D_i$ and $N_i$ are polinomial. My ...
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(reference-request) Theorem of Frobenius regarding transforming integer-valued matrix in diagonal matrix with divisibility conditions

A contest paper cites the following theorem as a theorem of Frobenius: Let $A$ be a $m \times n$ integer-valued matrix. There exists a positive integer $r \leq \min(m,n)$ and two unimodular matrices $...
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Find explicitely an element of SL$_2(\mathbb Z)$

Let $\Gamma=\text{SL}_2(\mathbb Z)$ act on $\mathbb H=\{z \in \mathbb C: Im(z) > 0\}$ by $\gamma z=\frac{az+b}{cz+d}$, where $\gamma$ is the matrix $\left[\begin{matrix} a & b \\ c & d \end{...
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609 views

If matrix $A$ is totally unimodular, then matrix $\begin{bmatrix} A &\pm A\end{bmatrix}$ is also totally unimodular

Given a totally unimodular matrix $A \in\{-1,0,1\}^{m\times n}$, show that the matrix $$\begin{bmatrix} A &\pm A\end{bmatrix}$$ is also totally unimodular. I want to prove that exchanging any two ...
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188 views

Is this block matrix totally unimodular?

Suppose matrix $A \in \mathbb{R}^{m×n}$ has the consecutive ones property and, thus, is totally unimodular. Is the following block matrix also totally unimodular (TU)? $$B = \begin{pmatrix} A ...
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Does this property imply total unimodularity?

I have a matrix $M$ such that every entry in $M$ is $1,-1$ or $0$. My matrix also has the following property. Let $v_i$ and $v_j$ be two columns of $M$, and let $S$ be the set of row indices $s$ such ...
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Number of conjugacy classes for the modular group

What is the number $N$ of conjugacy classes for the modular group (integer matrices with determinant one), corresponding to each trace T? Is there a reasonably closed formula that would give such ...
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Eigenvectors of PSL(2,Z) in terms of quadratic integers

Suppose I have a matrix A from the modular group, whose eigenvalues are numbers from a suitable quadratic field $Q(\sqrt D)$, where D depends on the trace of the matrix. Is there a way to find the ...
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Is there an alorithm for finding conjugating elements in the modular group?

Suppose I have two elements A, B of the modular group (i.e. 2x2 integer matrices with unit determinant) that have the same trace. I would like to know when these two matrices are conjugates of each ...
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383 views

Prove that a certain block combination of totally unimodular matrices is totally unimodular

I conjecture the following. Given three rectangular matrices $A$, $B$ and $C$ such that the following two block matrices $ \begin{bmatrix} A & B \\ \end{bmatrix} $ and $ \begin{bmatrix} ...
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Show that if $f_1(1+xy+x^4)+f_2(y^2+x-1)+f_3(xy-1)=1$ in $k[x,y]$ no $f_i$ can be a nonzero element of $k$.

This is a statement in "Using Algebraic Geometry" by Cox et. al., Exercise 27 of Section 5.1: Let $R=k[x_1,\dots,x_n]$. There exists unimodular rows $A$ with the property that no $\vec{f}\in R^l$ ...
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226 views

On the definition of totally unimodular matrix

I'm a bit confused about the definition of a totally unimodular matrix, since my lecture notes states that this matrix is not totally unimodular: $$\begin{pmatrix} 1 && 0 \\ 1 && -1 \...
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Unimodular matrices $M_{E_8}$ and $M_{\operatorname{Spin}(32)/\mathbb{Z}_2}$

Consider two unimodular matrices $M_{E_8}$ and $M_{\operatorname{Spin}(32)/\mathbb{Z}_2}$: $$M_{E_8} = \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \cr -1 & 2 & ...
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194 views

Totally unimodular matrix in norm minimization

Let's say we have $l_1$ norm minimization $||Ax-b||_1$ with matrix $A$ with consecutive ones property. Now, we can construct a mathematical formulation to solve this problem: \begin{align} \text{min} ...
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328 views

When is the adjacency matrix of a simple undirected graph totally unimodular?

I would like to know the graphs for which the (node-node) adjacency matrix is totally unimodular. Is the following true? The adjacency matrix of G is totally unimodular if and only if G is a tree.
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Is this block matrix also totally unimodular?

Suppose matrix $A\in \mathbb R^{m\times n}$ is totally unimodular (TUM). Is the following matrix also TUM? $$ \begin{pmatrix} A & 0 & 0 \\ 0 & A & 0 \\ ...
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299 views

Can we have an integer polytope when $A$ is not totally unimodular and $b$ is an integer vector?

I am kind of confused. I know the theorem that: " Let $A$ be totally unimodular and b an integer vector. The polytope $P :$= {$x$ | $Ax$ ≤ $b$} is integer (all vertices are integer)." So I'm ...
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Consider the matrix {1, -1; 1 -1}. If I multiply first row by -1, it loses the total unimodular property. [closed]

Seymor says that multiplying a row by -1 preserves total unimodularity. Please explain this fallacy.
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On the distribution of unimodular matrices generated by the Hermite normal form

A problem I'm currently considering requires me to generate (pseudo-)random Gaussian integer matrices with Gaussian integer matrix inverses. By analogy with an algorithm I know for generating random ...
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Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not ...