Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

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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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Total Uses: 125
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Publisher Info:
Publisher Name: University of North Texas
Place of Publication: Denton, Texas
Date(s):
  • Creation: December 2009
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:
Level: Doctoral
Discipline: Mathematics
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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:
  • OCLC: 666859281 |
  • UNTCAT: b3866635 |
  • ARK: ark:/67531/metadc28376
Resource Type: Thesis or Dissertation
Format: Text
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Access: Use restricted to UNT Community
License: Copyright
Holder: Bajracharya, Neeraj
Statement: Copyright is held by the author, unless otherwise noted. All rights reserved.