Insegnamento a.a. 2024-2025

Topics in linear algebra:

1. Review of basic concepts:

• Real and Complex vector spaces
• Linear operators
• Eigenvalues, eigenvectors

2. The Geometry of linear algebra:

• Real and Hermitian inner product
• Orthogonality, orthonormal bases
• Projections, orthogonal projections, variational characterization
• The Unitary and Orthogonal Group

3. Matrix norms

• Norms, p-norms
• Operator norms
• Hilbert-Schmidt inner product
• Spectral radius, Gelfand formula, variational characterization of the spectral radius*

 4. Hermitian and Normal operators

• Hermitian operators
• Normal operators
• Spectral Theorem
• Variational characterization of eigenvalues, Rayleigh quotient, generalized Rayleigh quotient
• Spectral mapping theorem
• Complements*: Infinite dimensional spaces, compact operators, Laplacian and Poincarè inequality

 5. SVD Decompositiona and applications

• Singular Values, variational characterization
• SVD decomposotion
• Eckart-Young, the closest rank k-matrix 
• Least square method, pseudo inverse

6. Multilinear Algebra*

• Multilinear forms
• Exterior product, exterior algebra
• Determinant, volume
• Cross-product in R^3, applications, and relations with projections

Topics in discrete mathematics:

1. Review of basic concepts

• Set theory
• Functions and relations
• Counting methods

2. Graph theory

• Review of the basics: Terminology, type of graphs, connectivity
• Representation of a graph: Adjacency matrix, Incidence matrix
• Spectral graph theory: The Laplacian matrix and its eigenvalues
• Cheeger inequalities*

3. More on counting*

• Generating functions
• Recurrence relations

4. Asymptotics

• Basics of asymptotic analysis: big O notation, hierarchy
• Big O manipulation, examples
• Asymptotic combinatorics: examples

• 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 graph spectral theory.

• 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 exterior products, inner products, trace operators, and operator functions.

• Solve counting problems using and determine the asymptotic growth in recurrence relations and combinatorial problems.

   

   

   

   

   

• 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