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Matrices whose hermitian part is positive definite


Johnson, Charles Royal (1972) Matrices whose hermitian part is positive definite. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZXNF-SB10.


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, 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 ∏n.

Powers are investigated and it is found that the only matrices A for which Am 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) = 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-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 -1A*.

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):
  • Taussky-Todd, Olga
Thesis Committee:
  • Unknown, Unknown
Defense Date:31 March 1972
Funding AgencyGrant Number
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:04182016-160159252
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9680
Deposited On:21 Apr 2016 15:21
Last Modified:09 Nov 2022 19:20

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