# Linear spanning sets for matrix spaces

Micheli, Giacomo; Rosenthal, Joachim; Vettori, Paolo (2015). Linear spanning sets for matrix spaces. Linear Algebra and its Applications, 483:309-322.

## Abstract

Necessary and sufficient conditions are given on matrices $\mathit{A}$, $\mathit{B}$ and $\mathit{S}$, having entries in some field $\mathbb{F}$ and suitable dimensions, such that the linear span of the terms $\mathit{A}^i \mathit{SB}^j$ over $\mathbb{F}$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $\mathbb{F}[\mathit{A}]\mathit{S} \mathbb{F}[\mathit{B}]$ when $\mathbb{F}$ is a $\mathbf{finite}$ field.

Necessary and sufficient conditions are given on matrices $\mathit{A}$, $\mathit{B}$ and $\mathit{S}$, having entries in some field $\mathbb{F}$ and suitable dimensions, such that the linear span of the terms $\mathit{A}^i \mathit{SB}^j$ over $\mathbb{F}$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $\mathbb{F}[\mathit{A}]\mathit{S} \mathbb{F}[\mathit{B}]$ when $\mathbb{F}$ is a $\mathbf{finite}$ field.

## Altmetrics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 26 June 2015 04 Feb 2016 09:59 05 Apr 2016 19:51 Elsevier 0024-3795 Publisher DOI. An embargo period may apply. https://doi.org/10.1016/j.laa.2015.06.008
Permanent URL: https://doi.org/10.5167/uzh-118595

Content: Accepted Version
Language: English
Filetype: PDF - Registered users only until 25 June 2017
Size: 191kB
View at publisher
Embargo till: 2017-06-25

## TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.