MATH 602 :
Mathematical Principles in Science, Part II
(WINTER 2008, 9:30am MWF, Instructor: Gerlach)
(WINTER 2008, 2:30pm MWF, Instructor: Tian)
 
 

KEY COURSE TOPICS

I. EIGENVALUES AND EIGENVECTORS

Adjoint of an operator
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
II. INFINITE DIMENSIONAL VECTOR SPACES: EXAMPLES
Sturm-Liouville systems: regular, periodic, and singular
Sturm-Liouville series
III. INFINITE DIMENSIONAL VECTOR SPACES: PRINCIPLES
Inner product spaces
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
Texts:(1) G. Strang: Linear Algebra and its Applications (Chapter 5,6, and Appendix A)
          (2) U.H. Gerlach: Linear Mathematics in Infinite Dimensions (Chapter 1,3)