Dissertation
A new formulation and uniqueness of solutions to A. Horns problem
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2012
DOI:
https://doi.org/10.17918/etd-4307
Abstract
This Ph. D thesis concerns A. Horns problem characterizing those triples ( ; ; ) forwhich there exist Hermitian matrices A;B;C with eigenvalues ; and respectively,satisfyingA + B = C:We will provide a new formulation of solutions to A. Horns problem by whichone can verify if a triple ( ; ; ) is a solution to A. Horns problem without the recursiveprocess to derive all Horns inequalities. Moreover, an extended result to thequestion what is the set of all possible eigenvalues of the sum of Hermitian matricesA1;A2; : : : ;An which have eigenvalues 1; 2; : : : ; n respectively is included. We alsostudy the question when the solution matrices A;B and C are unique up to simultaneousunitary similarity. This problem was solved by Knutson and Tao througha heavy algebraic geometry machine and we will prove two special cases by linearalgebra techniques. In addition, we will provide a more detailed proof for the case ; and lie in Qn+: We prove that uniqueness occurs exactly when there exists aunique hive with border ( ; ; ): When ; and are partitions, this means thatuniqueness occurs exactly when the Littlewood-Richardson coefficient C equals 1.
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Details
- Title
- A new formulation and uniqueness of solutions to A. Horns problem
- Creators
- Lei Cao - DU
- Contributors
- Hugo J. Woerdeman (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 4307; 991014632338804721