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13 votes
2 answers
697 views

in search of a transformation between determinants

Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$. I can show $\det A_n=\det B_n=2^{\...
T. Amdeberhan's user avatar
8 votes
0 answers
335 views

How to check two matrices for similitude over $\mathbb{Z}$?

General question. Let $A$ and $B$ be two $n\times n$-matrices over $\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar (i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)? ...
darij grinberg's user avatar
3 votes
1 answer
240 views

Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$

Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
Bipolar Minds's user avatar
2 votes
0 answers
90 views

decidability special case of column generation problem

I have the following problem: Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$ Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
Armin Weiß's user avatar
1 vote
0 answers
158 views

Comparing the volume of a rational lagrangian under a linear symplectomorphism.

Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace $...