(WINTER 2008, 9:30am MWF, Instructor: Gerlach)

(WINTER 2008, 2:30pm MWF, Instructor: Tian)

KEY COURSE TOPICS

I. EIGENVALUES AND EIGENVECTORS

Adjoint of an operatorII. INFINITE DIMENSIONAL VECTOR SPACES: EXAMPLES

Hermetian operators

Spectral theorem

Triangularization via unitary similarity transformation

Diagonalization of normal matrices

Positive definite matrices

Quadratic forms and the generalized eigenvalue problem

Extremization with linear constraints

Rayleigh quotient

Singular value decomposition of a rectangular matrix

Pseudo-inverse of a recangular matrix

Sturm-Liouville systems: regular, periodic, and singularIII. INFINITE DIMENSIONAL VECTOR SPACES: PRINCIPLES

Sturm-Liouville series

Inner product spacesTexts:(1) G. Strang:

Complete metric spaces

Hilbert spaces

Square summable series and square integrable functions

Least squares approximation

Projection theorem

Generalized Fourier coefficients

Bessel's inequality, Parceval's equality and completeness

Unitary transformation between Hilbert spaces

(2) U.H. Gerlach: