Welcome to Wasin So's Ph.D. Thesis Page
In 1991, I finished the Ph. D.
thesis entitled "Exponential formulas and spectral indices" under the
supervision of Professor Robert C. Thompson.
Following is the abstract of the thesis.
The object of study is the matrix exponential. We are especially interested in a product of matrix exponentials.
Chapter 1 is a collection of basic and known facts about the matrix exponentials.
The only new result is a necessary condition on the singular values of a matrix and those of its exponential.
In Chapter 2, we study the CampbellBakerHausdorff formula, which expresses the product of two exponentials as the exponential of a "sum".
Special attention is given to the estimation of domains of absolute convergence for different presentations of this formula and we use the techniques of majorizing series.
For a product of matrix exponentials, we prove some special cases of three conjectured formulas in Chapter 3.
As consequences, we have complete proofs of these formulas for 2x2 matrices. These proofs are mainly matrixtheoretic.
Chapter 4 is devoted to investigating the possibility of some naive matrix exponential formulas.
Finally, as an application of matrix exponential formulas, three different types
of spectral indices are proved to be equivalent in Chapter 5.
Following are published papers resulted from
the thesis.
 Singular values of matrix exponentials, with R. C. Thompson, Linear and
Multilinear Algebra 47 (2000) 249255.
  Some explicit formulas for the matrix exponential, with Dennis S.
Bernstein, IEEE Transaction Automatic Control 38 (1993) 12281231.
  Equality cases in matrix exponential inequalities. SIAM Journal on
Matrix Analysis and Applications 13 (1992) 11541158.
  Products of exponentials of Hermitian and complex symmetric matrices, with
Robert C. Thompson, Linear and Multilinear Algebra 29 (1991)
225233.
  Some properties of the CampbellBakerHausdorff series, with Jane Day and
Robert C. Thompson, Linear and Multilinear Algebra 29 (1991)
207224.
  Convergence domains for the CampbellBakerHausdorff formula, with Morris
Newman and Robert C. Thompson. Linear and Multilinear Algebra 24
(1989) 301310.

Lectures related to the thesis.
This series of lectures is based on my Ph.D. thesis
"Exponential Formulas and Spectral Indices",
which was finished in UC Santa Barbara
under the supervision of Professor R.C. Thompson.
There are 10 lectures, each of which is featured a published article
coming out of the thesis.
In each lecture, background and history will be provided to understand
the research. Then results from the published article will be given.
Later work in the same direction will be discussed. Unsolved problems
will be posed as conjectures or open questions.
 Matrix exponentials and their computation
 ( with Dennis S. Bernstein )
Some explicit formulas for the matrix exponential
IEEE Transaction Automatic Control 38 (1993) 12281231

 Matrix exponential inequalities
 ( with R.C. Thompson )
Singular values of matrix exponentials,
Linear and Multilinear Algebra 47 (2000) 249255

 CampbellBakerHausdorff series
 ( with Morris Newman and Robert C. Thompson )
Convergence domains for the CampbellBakerHausdorff formula,
Linear and Multilinear Algebra 24 (1989) 301310

 Generalized CampbellBakerHausdorff series
 ( with Jane Day and Robert C. Thompson )
Some properties of the CampbellBakerHausdorff series,
Linear and Multilinear Algebra 29 (1991) 207224.

 Matrix exponential forumlas
 ( with Robert C. Thompson )
Products of exponentials of Hermitian and complex symmetric matrices,
Linear and Multilinear Algebra 29 (1991) 225233.

 Trace of products of matrix exponentials
 Equality cases in matrix exponential inequalities.
SIAM Journal on Matrix Analysis and Applications 13 (1992) 11541158.

 Limiting behaviors of products of matrix exponentials
 ( with Shmuel Friedland )
On the product of matrix exponentials,
Linear Algebra and Applications 196 (1993) 193206.

 Weyl's inequalities
 Commutativity and spectra of Hermitian matrices,
Linear Algebra and its Applications 212213 (1994) 121130.

 Interlacing inequalities
 ( with R. Horn and N. Rhee )
Eigenvalue Inequalities and Equalities,
Linear Algebra and its Applications 270 (1998) 2944.

 Horn's conjecture
 ( with J. Day and R.C. Thompson )
The spectrum of a Hermitian matrix sum,
Linear Algebra and its Applications 280 (1998) 289332.

