A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. This thesis contains problems about refined inertias and minimum ranks of sign patterns. The refined inertia of a square real matrix $B$, denoted $\ri(B)$, is the ordered $4$-tuple $(n_+(B), \ n_-(B), \ n_z(B), \ 2n_p(B))$, where $n_+(B)$ (resp., $n_-(B)$) is the number of eigenvalues of $B$ with positive (resp., negative) real part, $n_z(B)$ is the number of zero eigenvalues of $B$, and $2n_p(B)$ is the number of pure imaginary eigenvalues of $B$. The minimum rank (resp., rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of $\cal A$. First, we identify all minimal critical sets of inertias and refined inertias for full sign patterns of order 3. Then we characterize the star sign patterns of order $n\ge 5$ that require the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$, which is an important set for the onset of Hopf bifurcation in dynamical systems. Finally, we establish a direct connection between condensed $m \times n $ sign patterns and zero-nonzero patterns with minimum rank $r$ and $m$ point-$n$ hyperplane configurations in ${\mathbb R}^{r-1}$. Some results about the rational realizability of the minimum ranks of sign patterns or zero-nonzero patterns are obtained.

Cette recherche, menée dans le cadre d'un doctorat de sociologie sous la direction de Philippe Combessie, repose principalement sur une enquête de 24 mois dans des ateliers de création artistique installés dans la maison d’arrêt pour hommes du centre pénitentiaire de Marseille, que nous avons conjuguée à une enquête de 13 mois dans le centre éducatif fermé de Montfavet. Le travail engagé a surtout reposé sur la volonté d'examiner le maillage entre art, prison et société, considérant finalement que l'enfermement le plus visible pouvait, quand il devenait le point central d'une expérience artistique de collaboration entre le dedans et le dehors, permettre de mener une réflexion sur d'autres types de cloisonnements à l’oeuvre dans la société. Prenant appui sur un dispositif de création collaborative, développé avec des groupes de participants incarcérés et de la société civile, nous nous sommes questionnée sur son sens, ses fonctions, ses effets, à trois niveaux : individuel, institutionnel et sociétal. Nous avons envisagé cette action artistique comme un terrain d’expérimentation de transformations multiples et avons élaboré comme point d’appui conceptuel la notion de triple inertie. Il nous fallait alors tenter une analyse qui ne compartimente pas les impacts en fonction des différents champs de répercussion, s'appuyant sur une perspective d'analyse des agencements préexistants et des réagencements produits. C'est ainsi qu'a émergé une problématique centrale, constituant une armature pour structurer l'ensemble des questions que nous souhaitions abordées dans ce travail, et qui peut ainsi se formuler : en quoi l'expérience artistique collaborative en prison, dans une dynamique transformative, constitue une mise en mouvement et un dépassement des inerties et frontières carcérales à trois niveaux indissociables, individuel, institutionnel et sociétal? / This research, led for the purpose of a PhD in sociology under the supervision of Philippe Combessie, is based on a 24 months long study in the artistic experience workshops set up inside the Marseille correctional institution, combined with a 13 months study in Montfavet’s closed educative center. The work that has been undertaken rested upon the ability to examine the links between, art, prison and society, arguing that the most visible imprisonment can, when it becomes the central point of an artistic experience of collaboration between the inside and the outside, give the possibility to think of new forms of confinement at work in society. This project was developed according to an apparatus of collaborative creation established with various groups of participants that were either in custody or coming from civil society. Its meaning, its functions, and its effects, were questioned on three levels: individual, institutional, and societal. This artistic action came about as a space of experimentation for various transformations and gave the possibility to coin the notion of triple inertia, hereby used as conceptual tool. It was important to bring about an analysis that did not divide the impacts in terms of the various fields of repercussion, and instead, relied on an analysis of both pre-existing layouts, and the new forms of layouts produced. That is how the central issue emerged, becoming the framework that would structure the totality of the questions that were to be discussed in this work, and that can be stipulated as such: How does the collaborative artistic experience in prison, through a transformative dynamic, constitute a movement as well as an overcoming of inertias and prison frontiers on three inseparable levels, individual, institutional and societal.