Abstract in English / The research will analyze and compare the current research on the spectral
factorization of non-singular and singular matrices. We show that a nonsingular non-scalar matrix A can be written as a product A = BC where the eigenvalues of B and C are arbitrarily prescribed subject to the condition that the product of the eigenvalues of B and C must be equal to the determinant of A. Further, B and C can be simultaneously triangularised as a lower and upper triangular matrix respectively. Singular matrices will
be factorized in terms of nilpotent matrices and otherwise over an arbitrary
or complex field in order to present an integrated and detailed report on the
current state of research in this area. Applications related to unipotent, positive-definite, commutator, involutory and Hermitian factorization are studied for non-singular matrices, while applications related to positive-semidefinite matrices are investigated for singular matrices. We will consider the theorems found in Sourour [24] and Laffey [17] to show
that a non-singular non-scalar matrix can be factorized spectrally. The same
two articles will be used to show applications to unipotent, positive-definite
and commutator factorization. Applications related to Hermitian factorization will be considered in [26]. Laffey [18] shows that a non-singular matrix
A with det A = ±1 is a product of four involutions with certain conditions
on the arbitrary field. To aid with this conclusion a thorough study is made
of Hoffman [13], who shows that an invertible linear transformation T of a
finite dimensional vector space over a field is a product of two involutions
if and only if T is similar to T−1. Sourour shows in [24] that if A is an
n × n matrix over an arbitrary field containing at least n + 2 elements and
if det A = ±1, then A is the product of at most four involutions.
We will review the work of Wu [29] and show that a singular matrix A of
order n ≥ 2 over the complex field can be expressed as a product of two
nilpotent matrices, where the rank of each of the factors is the same as A,
except when A is a 2 × 2 nilpotent matrix of rank one.
Nilpotent factorization of singular matrices over an arbitrary field will also
be investigated. Laffey [17] shows that the result of Wu, which he established
over the complex field, is also valid over an arbitrary field by making use
of a special matrix factorization involving similarity to an LU factorization.
His proof is based on an application of Fitting's Lemma to express, up to
similarity, a singular matrix as a direct sum of a non-singular and nilpotent matrix, and then to write the non-singular component as a product of a lower and upper triangular matrix using a matrix factorization theorem of Sourour [24]. The main theorem by Sourour and Tang [26] will be investigated to highlight the necessary and sufficient conditions for a singular matrix to be written as a product of two matrices with prescribed eigenvalues. This result is used to prove applications related to positive-semidefinite matrices for singular matrices. / National Research Foundation of South Africa / Mathematical Sciences / M Sc. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/26844 |
| Date | 06 1900 |
| Creators | Gaoseb, Frans Otto |
| Contributors | Botha, J.D. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | 1 online resource (94 leaves), application/pdf |
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