Insegnamento a.a. 2025-2026

Topics in Linear Algebra:

Review of Basic Concepts:

• Real and Complex vector spaces

• Linear operators

• Eigenvalues, eigenvectors

The Geometry of Linear Algebra:

• Real and Hermitian inner product

• Orthogonality, orthonormal bases

• Projections, orthogonal projections, variational characterization

• The Unitary and Orthogonal Group

Matrix Norms

• Norms, p-norms

• Operator norms

• Hilbert-Schmidt inner product

• Spectral radius, Gelfand formula, variational characterization of the spectral radius

Hermitian and Normal Operators

• Hermitian operators

• Normal operators

• Spectral Theorem

• Variational characterization of eigenvalues, Rayleigh quotient, generalized Rayleigh quotient

• Spectral mapping theorem*

SVD Decomposition and Applications

• Singular values, variational characterization

• SVD decomposition

• Eckart-Young, the closest rank-k matrix

• Least squares method, pseudo-inverse

Applications in Computing and Modeling:

Applications to Graph Theory

• Review of the basics: Terminology, types of graphs, connectivity

• Representation of a graph: Adjacency matrix, Incidence matrix

• Spectral graph theory: The Laplacian matrix and its eigenvalues

• Applications to network analysis*

Perron-Frobenius Theory

• The spectrum of positive matrices

• Stochastic, bistochastic matrices

• Ergodicity of Markov chains

DeGroot Learning

PageRank Algorithm

• Comprehend the geometric concepts involved in the formalism of linear algebra.
• Understand scalar products, norms, and operator norms in real and complex settings.
• Develop familiarity with the spectral theorem for normal and Hermitian operators.
• Understand the SVD decomposition and its applications.
• Acquire a basic understanding of Perron-Frobenius theory and its applications

• Perform diagonalization and matrix decompositions, both real and complex, and utilize these tools to compute projections and matrix functions.

• Use scalar products, norms, and operator norms in optimization and variational problems.

• Manipulate expressions involving inner products, trace operators, and operator functions.

• Use linear algebra tools to model and analyze systems in computer science, such as networks, search algorithms, and consensus dynamics.

• Practical Exercises

Students are assigned weekly exercises that are directly related to the concepts taught during the week. These exercises are meant to be solved independently by the students.

During the subsequent week's class, we will collaboratively solve and engage in discussions on the assigned exercises.

This approach promotes active learning, as students have the opportunity to engage in collaborative problem-solving. By discussing the exercises, students can clarify any misconceptions, deepen their understanding, and learn alternative approaches to problem-solving.

• Partial exams consist of two written exams, one at the midpoint of the course and one at the end. The final grade is calculated as the average of these two scores.
• General exam: written exam at the end of the course that contributes to the overall assessment.

Each written exam comprises five exercises with multiple bullet points covering all the presented material.

A total of 34 points will be assigned, with scores above 34 receiving the maximum grade. The exam evaluates the acquisition of basic knowledge and problem-solving abilities. The exercises vary in difficulty, with the first three emphasizing fundamental concepts and accounting for 25 out of 34 points, while the last two require more problem-solving and critical thinking, accounting for the remaining points.

• Lecture notes of the course
• Introduction to linear algebra. Gilbert Strang
• Linear algebra and learning from data. Gilbert Strang
• Concrete Mathematics. Ronald L. Graham, Donald E. Knuth, Oren Patashnik