# 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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### Proof of Hoffman & Kruskal's theorem on Unimodularity and Integrality.

I'm reading the following proof. Thm. (Unimodularity and Integrality, Hoffman & Kruskal ): Let $A \in \mathbb{R}^{m \times n}, \operatorname{rank} A=n .$ The following are equivalent: a) $A$...
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### 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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### 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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### 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 ...