Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Irreducible Matrices and Strong Connectivity

In course

ME-427: Networked control systems

This course offers an introduction to control systems using communication networks for interfacing sensors, actuators, controllers, and processes. Challenges due to network non-idealities and opportunities offered by communication will be analyzed. ...

Description

This lecture covers the concepts of irreducible matrices and strong connectivity in networked control systems. It discusses the properties of adjacency matrices, primitive matrices, and periodicity in digraphs. The instructor explains the implications of powers of matrices, the relationship between irreducibility and strong connectivity, and the interpretation of graph structures in the context of matrix theory.

Instructor

Giancarlo Ferrari Trecate

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Algebra: Linear algebra

Related concepts (32)

Adjacency matrix

In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Graph (abstract data type)

In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as edges but also sometimes arrows or arcs.

Adjacency list

In graph theory and computer science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the set of neighbors of a particular vertex in the graph. This is one of several commonly used representations of graphs for use in computer programs. An adjacency list representation for a graph associates each vertex in the graph with the collection of its neighbouring vertices or edges.

Graph (discrete mathematics)

In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.

Related lectures (41)

Networked Control Systems: Properties and Connectivity

Explores properties of matrices, irreducibility, and graph connectivity in networked control systems.

Graphical Models: Representing Probabilistic Distributions

Covers graphical models for probabilistic distributions using graphs, nodes, and edges.

Consensus in Networked Control Systems

Explores consensus in networked control systems through graph weight design and matrix properties.

Laplacian Matrix: Properties and Examples

Explores the Laplacian matrix, time-varying consensus theorems, and balanced graphs in networked control systems.

Graph Algorithms: Modeling and Traversal