MATH487, Fall 2018: class plan


Introduction and course overview

Systems of linear equations.
Gauss elimination


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.

Determinants: definition.


Determinant: existence and properties.
Matrix is singular iff its determinant vanishes.
Determinants and \(LU\) decomposition.
Example: tridiagonal matrices.
Kramer’s rule.


Remarkable determinants: VanderMonde, Cauchy.
Applications: Lagrange interpolation formula.
Minors, adjoint matrices.


Cauchy-Binet formula and matrix tree theorem.


General linear spaces.
Linear operators.
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.


Matrix exponentials.
Shift-invariant spaces of functions, linear ODEs.


Conjugate spaces.
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.


Midterm 1


Applications: low rank approximations (Netflix problem).

Week of 10.8

Rotation matrices in 3D spaces, SO(3).
Roll, pitch, yaw. Commutators; Jacobi identity.

Vector calculus.
Classical identities in 3D; scalar, vector products.

Line integrals.
Integrals over higher-dimensional chains.
Exterior forms.

Week of 10.15

Differential forms.
Stokes and relatives.
Applications: data analysis.
Div, grad, rot. Maxwell equations.

Week of 10.22

Metric spaces: completeness, examples.
Contractions and their fixpoints.

Hyperbolic spaces.
Applications: search and navigation.

Complex analysis.
Analytic functions. Cauchy-Riemann.
Laurent series. Domains of convergence.
Integrals via residues.

Week of 10. 29

Logarithmic residues, Rouche theorem.
Applications: Ideal fluids.
Fractional-linear mappings.

Midterm 2

Week of 11.5

Elements of Fourier transform.
Parseval theorem.
Inverse transform.
Duality (smoothness-localization). “Uncertainty principle”.

Week of 11.12.

Elements of asymptotic analysis.
Laplace method. Steepest descent.
Tauber theorems.
Asymptotics for coefficients of generating functions; Darboux transfer.

Week of 11.19

no classes

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.
Elliptic problems.

Week of 12.10

Harmonic functions.
Dirichlet, Neumann.

Course recap.