Citation
Uhlig, Frank Detlev (1972) A Study of the Canonical Form for a Pair of Real Symmetric Matrices and Applications to Pencils and to Pairs of Quadratic Forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/pfpv-ty08. https://resolver.caltech.edu/CaltechTHESIS:06272025-174950440
Abstract
A pair of real symmetric matrices S and T is called a nonsingular pair if Sis nonsingular. A new treatment for obtaining the classical canonical pair form for a nonsingular pair is obtained by the use of results on commuting matrices and by elementary matrix algebra. This canonical form is used to obtain formulas for an arbitrary real n X n matrix A that relate the dimensions of both the space N of real I symmetric matrices T such that AT = TA and the space of products AT such that AT is symmetric to the real Jordan normal form of A. The first formula expresses a previously found result in a simpler way while the second one is new. These formulas are then applied to prove anew the known result that A is nonderogatory iff dim N = n. Simultaneous diagonalization of two real symmetric matrices has been of interest. For instance it has been shown that if the quadratic forms associated with Sand T (of dimensions greater than 2) do not vanish simultaneously, then S and T can be diagonalized simultaneously by a real congruence transformation. This subject is generalized here to the study of the following two problems:
1) The finest simultaneous block diagonal structure for nonsingular pairs,
2) common annihilating vectors of the corresponding quadratic forms. The proofs are obtained here by algebraic means. Results: ad 1) A simultaneous block diagonalization X' TX= diag(A1,...,Ak and X'TX = diag(B1,...,Bk) with dim Ai = dim Bi and X nonsingular is the finest simultaneous block diagonalization of a nonsingular pair S and T, if k is maximal. In this finest diagonalization the sizes of the blocks Ai are uniquely determined (up to permutations) by any set of generators of the pencil P(S,T) = {aS + bT]a,b ϵ R}. The number k and the sizes of the diagonal blocks are also derived from the factorization over C of f(λ,µ) = det(λS + µT) for λ, µ ϵ R. ad 2) Knowing the real Jordan normal form of S-1T for a nonsingular pair S and T we compute the maximal number m of linearly independent vectors that are simultaneously annihilated by the corresponding quadratic forms. Conversely, knowing m for two quadratic forms we deduce the first simultaneous block diagonal structure of S and T, the corresponding pair of real symmetric matrices. This is used to give new sufficient conditions for S and T to be simultaneously diagonalizable.
Item Type: | Thesis (Dissertation (Ph.D.)) | ||||
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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): |
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Thesis Committee: |
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Defense Date: | 9 November 1971 | ||||
Record Number: | CaltechTHESIS:06272025-174950440 | ||||
Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:06272025-174950440 | ||||
DOI: | 10.7907/pfpv-ty08 | ||||
ORCID: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||
ID Code: | 17494 | ||||
Collection: | CaltechTHESIS | ||||
Deposited By: | Benjamin Perez | ||||
Deposited On: | 27 Jun 2025 21:49 | ||||
Last Modified: | 27 Jun 2025 22:20 |
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