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

Access: Use of this item is restricted to the UNT Community 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. Bajracharya, Neeraj December 2009 UNT Libraries UNT Theses and Dissertations Total Uses: 147 Past 30 days: 0 Yesterday: 0
Creator (Author): Publisher Name: University of North Texas Place of Publication: Denton, Texas Creation: December 2009 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. Name: Doctor of Philosophy Level: Doctoral Discipline: Mathematics Department: Department of Mathematics Grantor: University of North Texas LCSH : Angles (Geometry) LCSH : Curves. LCSH : Matrices. Jordan product | Eigenvalues | angle function of a matrix | symmetric matrix | positive definite matrix | level curve Chair : Conley, Charles Major Professor Committee Member : Mauldin, R. Daniel Committee Member : Cherry, William A. UNT Libraries UNT Theses and Dissertations OCLC: 666859281 | UNTCAT: b3866635 | ARK: ark:/67531/metadc28376 Thesis or Dissertation Text Access: Use restricted to UNT Community License: Copyright Holder: Bajracharya, Neeraj Statement: Copyright is held by the author, unless otherwise noted. All rights reserved.