Phd records
https://feeds.library.caltech.edu/people/Johnson-Charles-Royal/Phd.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:15:46 +0000Matrices whose hermitian part is positive definite
https://resolver.caltech.edu/CaltechTHESIS:04182016-160159252
Authors: {'items': [{'id': 'Johnson-Charles-Royal', 'name': {'family': 'Johnson', 'given': 'Charles Royal'}}]}
Year: 1972
DOI: 10.7907/ZXNF-SB10
<p>We are concerned with the class ∏<sub>n</sub> of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite.</p>
<p>Various connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diagonal entries, then there is a positive diagonal F such that FA ϵ ∏<sub>n</sub>. </p>
<p>Powers are investigated and it is found that the only matrices A for which A<sup>m</sup> ϵ ∏<sub>n</sub> for all integers m are the Hermitian elements of ∏<sub>n</sub>. Products and sums are considered and criteria are developed for AB to be in ∏<sub>n</sub>.</p>
<p>Since ∏<sub>n</sub> n is closed under inversion, relations between H(A)<sup>-1</sup> and H(A<sup>-1</sup>) are studied and a dichotomy observed between the real and complex cases. In the real case more can be said and the initial result is that for A ϵ ∏<sub>n</sub>, the difference H(adjA) - adjH(A) ≥ 0 always and is ˃ 0 if and only if S(A) = A-A*/2 has more than one pair of conjugate non-zero characteristic roots. This is refined to characterize real c for
which cH(A<sup>-1</sup>) - H(A)<sup>-1</sup> is positive definite.</p>
<p>The cramped (characteristic roots on an arc of less than 180°) unitary matrices are linked to ∏<sub>n</sub> and characterized in several ways via products of the form A <sup>-1</sup>A*.</p>
<p>Classical inequalities for Hermitian positive definite matrices are studied in ∏<sub>n</sub> and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of ∏<sub>n</sub> in which the precise statement of Hadamardis inequality holds is isolated while in another large subclass its reverse is shown to hold. In the second Hadamard's inequality is weakened in such a way that it holds throughout ∏<sub>n</sub>. Both approaches contain the original Hadamard inequality as a special case. </p>
https://thesis.library.caltech.edu/id/eprint/9680