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Description:  Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix. 
Creator(s):  Bajracharya, Neeraj 
Creation Date:  December 2009 
Partner(s): 
UNT Libraries

Collection(s): 
UNT Theses and Dissertations

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Publisher Name: University of North Texas
Place of Publication: Denton, Texas


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Description:  Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix. 

Degree: 
Name:
Doctor of Philosophy
Level:
Doctoral
Discipline:
Mathematics
Department:
Department of Mathematics
Grantor:
University of North Texas


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Subject(s): 


Keyword(s):  Jordan product  Eigenvalues  angle function of a matrix  symmetric matrix  positive definite matrix  level curve  
Contributor(s): 


Partner: 
UNT Libraries


Collection: 
UNT Theses and Dissertations


Identifier:  
Resource Type:  Thesis or Dissertation  
Format:  Text  
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Use restricted to UNT Community
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Copyright
Holder:
Bajracharya, Neeraj
Statement:
Copyright is held by the author, unless otherwise noted. All rights reserved.
