Citation
Johnson, Charles Royal (1972) Matrices whose hermitian part is positive definite. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZXNFSB10. https://resolver.caltech.edu/CaltechTHESIS:04182016160159252
Abstract
We are concerned with the class ∏_{n} of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite.
Various connections are established with other classes such as the stable, Dstable 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 ∏_{n}.
Powers are investigated and it is found that the only matrices A for which A^{m} ϵ ∏_{n} for all integers m are the Hermitian elements of ∏_{n}. Products and sums are considered and criteria are developed for AB to be in ∏_{n}.
Since ∏_{n} n is closed under inversion, relations between H(A)^{1} and H(A^{1}) 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 ∏_{n}, the difference H(adjA)  adjH(A) ≥ 0 always and is ˃ 0 if and only if S(A) = AA*/2 has more than one pair of conjugate nonzero characteristic roots. This is refined to characterize real c for which cH(A^{1})  H(A)^{1} is positive definite.
The cramped (characteristic roots on an arc of less than 180°) unitary matrices are linked to ∏_{n} and characterized in several ways via products of the form A ^{1}A*.
Classical inequalities for Hermitian positive definite matrices are studied in ∏_{n} and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of ∏_{n} 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 ∏_{n}. Both approaches contain the original Hadamard inequality as a special case.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  31 March 1972  
Funders: 
 
Record Number:  CaltechTHESIS:04182016160159252  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:04182016160159252  
DOI:  10.7907/ZXNFSB10  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9680  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  21 Apr 2016 15:21  
Last Modified:  09 Nov 2022 19:20 
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