Dr. Hugo J. Woerdeman
hugo@math.drexel.edu
Korman 206
Office Hours: TBA

Class meeting times: Mondays, 6:00 pm - 08:50 pm

Matrix completion problems are concerned with finding a matrix of which some entries are prescribed and which belongs to a certain class. In this course we shall mainly focus on the class of positive definite matrices, as this class appears in many applications. As an example, the trigonometric moment problem, where a positive valued function defined on the unit circle in the complex plane is sought that has certain prescribed Fourier coefficients, may be viewed as a positive definite matrix completion problem. In this course we will treat these classical problems which have applications in filter design,
but we will also encounter some very recent results that appear in the active research area of multivariable moments problems.

Central to this course are the following ten results, which will be treated along with their background, consequences and applications:

1. Caratheodory's inetrpolation theorem
2. Szego's Theorem on stability of orthogonal polynomial
3. Grone-Johnson-Sa-Wolkowicz Theorem on positive semidefinite completions
4. Paulsen-Power-Smith's result on extreme rays of cones of positive semidefinite matrices
5. Gabardo/Bakonyi-Naevdal's result for multivariate moment problems
6. Geronimo-Woerdeman's result on the two variable autoregressive filter problem
7. Johnson-Rodman/Woerdeman's results on Hermitian completions
8. Krein's Theorem on orthogonal polynomials
9. Nehari's theorem and it's use in H-infinity control
10. Normal matrix completions and their appearance in Quantum Computation

The course assumes a basic knowledge of linear algebra, including: matrix algebraic operations, vector spaces, rank, row/column space, null space, orthogonality, determinants and eigenvalues and eigenvectors, Hermitian and positive definite matrices. In addition, some basic knowledge of analysis is helpful, including continuity, open and closed sets, compactness, convergence of series. A key to success in this course is to be extremely familiar with the multiplication of matrices, including partitioned matrices. Finally, all vector spaces will be over the complex field, so familiarity with the complex field is assumed.

Matrix Analysis - by Roger A. Horn, Charles R. Johnson
Topics in Matrix Analysis - by Roger A. Horn, Charles R. Johnson
The Theory of Matrices - by Peter Lancaster, Miron Tismenetsky
Toeplitz Forms and Their Applications - by Ulf Grenander and Gabor Szego