Citation
Loewy, Raphael (1972) On the Lyapunov Transformation for Stable Matrices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/EXGT0968. https://resolver.caltech.edu/CaltechTHESIS:05092016130648083
Abstract
The matrices studied here are positive stable (or briefly stable). These are matrices, real or complex, whose eigenvalues have positive real parts. A theorem of Lyapunov states that A is stable if and only if there exists H ˃ 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation L_{A} :H → AH + HA* are discussed.
1. Let C_{1} (A) = {AH + HA* :H ≥ 0} and C_{2} (A) = {H: AH+HA* ≥ 0}. The problems of determining the cones C_{1}(A) and C_{2}(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C_{1}(A) is the polar of C_{2}(A*), and it is also shown that C_{1} (A) = C_{1}(A^{1}). The inertia assumed by matrices in C_{1}(A) is characterized.
2. The index of dissipation of A was defined to be the maximum number of equal eigenvalues of H, where H runs through all matrices in the interior of C_{2}(A). Upper and lower bounds, as well as some properties of this index, are given.
3. We consider the minimal eigenvalue of the Lyapunov transform AH+HA*, where H varies over the set of all positive semidefinite matrices whose largest eigenvalue is less than or equal to one. Denote it by ψ(A). It is proved that if A is Hermitian and has eigenvalues μ_{1} ≥ μ_{2}…≥ μ_{n} ˃ 0, then ψ(A) = (μ_{1}μ_{n})^{2}/(4(μ_{1} + μ_{n})). The value of ψ(A) is also determined in case A is a normal, stable matrix. Then ψ(A) can be expressed in terms of at most three of the eigenvalues of A. If A is an arbitrary stable matrix, then upper and lower bounds for ψ(A) are obtained.
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:  8 March 1972  
Funders: 
 
Record Number:  CaltechTHESIS:05092016130648083  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05092016130648083  
DOI:  10.7907/EXGT0968  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9710  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  09 May 2016 22:20  
Last Modified:  01 Jul 2024 21:52 
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