Introduction and course overview
Systems of linear equations.
Basic properties of matrices.
Regular matrices; \(LU\) decomposition of regular matrices.
Notion of (matrix) group. Examples: groups of lower, upper triangular matrices. Matrix inverses.
Row swaps and permutation matrices. Group of permutation matrices.
Non-singular matrices and row-echelon normal form.
Determinant: existence and properties.
Matrix is singular iff its determinant vanishes.
Determinants and \(LU\) decomposition.
Example: tridiagonal matrices.
Remarkable determinants: VanderMonde, Cauchy.
Applications: Lagrange interpolation formula.
Minors, adjoint matrices.
Cauchy-Binet formula and matrix tree theorem.
General linear spaces.
Bases. Bases changes.
Normal forms: mappings between different spaces: normal forms
Endomorphisms: classification problem.
Eigenvectors. Characteristic polynomial.
Applications: Markov chains; random walks; transition matrices; random walks on binary cubes and on cycles.
Reminder: complex numbers. Matrix interpretation.
Jordan normal forms.
Functions of matrices.
Shift-invariant spaces of functions, linear ODEs.
Mappings to conjugate spaces. Quadratic forms.
Normal forms for quadratic forms. Signature, inertia theorem.
Applications of quadratic forms:
Statistics; variance, correlations.
Orthogonalization. Orthogonal polynomials.
Self-adjoint operators and their spectra.
Rayleigh min-max theorem; spectra interlacing.
Applications: low rank approximations (Netflix problem).
Week of 10.8
Rotation matrices in 3D spaces, SO(3).
Roll, pitch, yaw. Commutators; Jacobi identity.
Classical identities in 3D; scalar, vector products.
Integrals over higher-dimensional chains.
Week of 10.15
Stokes and relatives.
Applications: data analysis.
Div, grad, rot. Maxwell equations.
Week of 10.22
Metric spaces: completeness, examples.
Contractions and their fixpoints.
Applications: search and navigation.
Analytic functions. Cauchy-Riemann.
Laurent series. Domains of convergence.
Integrals via residues.
Week of 10. 29
Logarithmic residues, Rouche theorem.
Applications: Ideal fluids.
Week of 11.5
Elements of Fourier transform.
Duality (smoothness-localization). “Uncertainty principle”.
Week of 11.12.
Elements of asymptotic analysis.
Laplace method. Steepest descent.
Asymptotics for coefficients of generating functions; Darboux transfer.
Week of 11.19
Week of 11.26
Partial differential equations.
First order equations; examples (eikonal).
Cauchy problem. Quasilinear PDEs.
Week of 12. 3
Wave equation: d’Alembert solution.
Wave equation via Fourier series.
Week of 12.10