Ballantine's theorem on Sp(2n, C)

Among normal matrices, positive definite matrices or positive semidefinite matrices are important. Any product of two positive definite (semidefinite) matrices is diagonalizable and has positive (nonnegative) eigenvalues so such products do not fill up Mn(C) or GL(n, C), the group of n × n nonsingul...

Full description

Saved in:
Bibliographic Details
Main Author: Granario, Daryl Q.
Format: text
Published: Animo Repository 2012
Subjects:
Online Access:https://animorepository.dlsu.edu.ph/faculty_research/11356
Tags: Add Tag
No Tags, Be the first to tag this record!
Institution: De La Salle University
id oai:animorepository.dlsu.edu.ph:faculty_research-13343
record_format eprints
spelling oai:animorepository.dlsu.edu.ph:faculty_research-133432023-12-01T23:38:27Z Ballantine's theorem on Sp(2n, C) Granario, Daryl Q. Among normal matrices, positive definite matrices or positive semidefinite matrices are important. Any product of two positive definite (semidefinite) matrices is diagonalizable and has positive (nonnegative) eigenvalues so such products do not fill up Mn(C) or GL(n, C), the group of n × n nonsingular matrices.. Ballantine [1, 2, 3, 4] showed that every matrix with positive de- terminant is a product of five positive definite matrices. Radjavi [11] showed that every matrix A with real determinant is a product of at most four Her- mitian matrices, and there are matrices which are not product of less. Wu [14] showed that every matrix with nonnegative determinant is a product of five positive semidefinite matrices. In the same paper, Wu also gave a char- acterization of matrices that can be expressed as a product of four positive semidefinite matrices. A recent work of Cui, Li, and Sze [5] gives a character- ization of products of three positive semidefinite matrices. We are interested in the following problem. Problem. Let G be a subgroup of GL(n, C) consisting of matrices with positive determinant. Can the matrices in G be written as a product of a finite number of positive definite matrices in G? If so, how many factors do we need to express all elements of G as products of positive definite matrices in G? In this talk, we consider the complex symplectic group Sp(2n, C): Sp(2n, C) = {A ∈ GL(2n, C) : ATJnA = Jn} , where Jn = [0 In] [−In 0] The symplectic group is a classical group defined as the set of linear trans- formations of a 2n-dimensional vector space over C which preserve the non- degenerate skew-symmetric bilinear form which is defined by Jn. It is a non-compact, simply connected, and simple Lie group [8]. Every symplectic matrix A ∈ Sp(2n, C) is nonsingular with the inverse A−1 = J −1 n A>Jn. It is obvious to see that det A = ±1 but less obvious to narrow it down to det A = 1 though it is true. So one may write Sp(2n, C) = {A ∈ SL(2n, C) : J −1 n A >Jn = A −1 }. One can see that Sp(2n, C) is invariant under ∗, the conjugate transpose. We drop the subscript if the size of the matrix Jn is clear from context. For A ∈ GL(2n, C), we define AJ = J−1ATJ, the J-adjoint of A. In block form, one can write Sp(2n, C) = {A B C D ∈ SL(2n, C) : A TC = C TA, B TD = DTB, ATD − CTB = In} . 2012-01-01T08:00:00Z text https://animorepository.dlsu.edu.ph/faculty_research/11356 Faculty Research Work Animo Repository Positive-definite functions Symplectic groups Mathematics
institution De La Salle University
building De La Salle University Library
continent Asia
country Philippines
Philippines
content_provider De La Salle University Library
collection DLSU Institutional Repository
topic Positive-definite functions
Symplectic groups
Mathematics
spellingShingle Positive-definite functions
Symplectic groups
Mathematics
Granario, Daryl Q.
Ballantine's theorem on Sp(2n, C)
description Among normal matrices, positive definite matrices or positive semidefinite matrices are important. Any product of two positive definite (semidefinite) matrices is diagonalizable and has positive (nonnegative) eigenvalues so such products do not fill up Mn(C) or GL(n, C), the group of n × n nonsingular matrices.. Ballantine [1, 2, 3, 4] showed that every matrix with positive de- terminant is a product of five positive definite matrices. Radjavi [11] showed that every matrix A with real determinant is a product of at most four Her- mitian matrices, and there are matrices which are not product of less. Wu [14] showed that every matrix with nonnegative determinant is a product of five positive semidefinite matrices. In the same paper, Wu also gave a char- acterization of matrices that can be expressed as a product of four positive semidefinite matrices. A recent work of Cui, Li, and Sze [5] gives a character- ization of products of three positive semidefinite matrices. We are interested in the following problem. Problem. Let G be a subgroup of GL(n, C) consisting of matrices with positive determinant. Can the matrices in G be written as a product of a finite number of positive definite matrices in G? If so, how many factors do we need to express all elements of G as products of positive definite matrices in G? In this talk, we consider the complex symplectic group Sp(2n, C): Sp(2n, C) = {A ∈ GL(2n, C) : ATJnA = Jn} , where Jn = [0 In] [−In 0] The symplectic group is a classical group defined as the set of linear trans- formations of a 2n-dimensional vector space over C which preserve the non- degenerate skew-symmetric bilinear form which is defined by Jn. It is a non-compact, simply connected, and simple Lie group [8]. Every symplectic matrix A ∈ Sp(2n, C) is nonsingular with the inverse A−1 = J −1 n A>Jn. It is obvious to see that det A = ±1 but less obvious to narrow it down to det A = 1 though it is true. So one may write Sp(2n, C) = {A ∈ SL(2n, C) : J −1 n A >Jn = A −1 }. One can see that Sp(2n, C) is invariant under ∗, the conjugate transpose. We drop the subscript if the size of the matrix Jn is clear from context. For A ∈ GL(2n, C), we define AJ = J−1ATJ, the J-adjoint of A. In block form, one can write Sp(2n, C) = {A B C D ∈ SL(2n, C) : A TC = C TA, B TD = DTB, ATD − CTB = In} .
format text
author Granario, Daryl Q.
author_facet Granario, Daryl Q.
author_sort Granario, Daryl Q.
title Ballantine's theorem on Sp(2n, C)
title_short Ballantine's theorem on Sp(2n, C)
title_full Ballantine's theorem on Sp(2n, C)
title_fullStr Ballantine's theorem on Sp(2n, C)
title_full_unstemmed Ballantine's theorem on Sp(2n, C)
title_sort ballantine's theorem on sp(2n, c)
publisher Animo Repository
publishDate 2012
url https://animorepository.dlsu.edu.ph/faculty_research/11356
_version_ 1784863533758939136