Questions tagged [smith-normal-form]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
110 views

Distribution of Smith normal forms for lower triangular matrices with given diagonals

Given integers $m$ and $n$ and $d_1, \ldots, d_m \in \mathbb{Z}/n \mathbb{Z}$, consider the set of all lower-triangular matrices of dimension $m$ with diagonal elements equal to $d_i$. What can be ...
9
votes
0answers
93 views

Smith normal form of conjugacy class actions

This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group. Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the symmetric group $S_n$. Identify a ...
8
votes
1answer
262 views

Smith Normal Form of a Cayley Graph of the Symmetric Group

Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
4
votes
1answer
132 views

Smith normal form and affine buildings

In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
8
votes
2answers
342 views

Generalized Smith Theorem for the torsion of cokernels

Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$. Let $I_k=(\det \...
8
votes
1answer
618 views

Computation time of Smith normal form in Maple

I am using Maple to compute the Smith normal form (SNF) of a $120 \times 120$ matrix and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is ...
19
votes
1answer
747 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
4
votes
1answer
409 views

Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers? In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...
15
votes
1answer
1k views

Smith Normal Form of powers of a matrix

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix? The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
4
votes
1answer
373 views

Smith Normal Form

Let $R=Z[x_{1},x_{2},\dots,x_{n}]$ be a multivariate polynomial ring. Is it possible to define a normal form for a general $m \times m$ matrix $M$ with entries from $R$?
27
votes
5answers
2k views

Does Smith normal form imply PID?

Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain? If this fails, suppose we ...