This thesis can be classified as being part of Matrix Analysis.
Within this area, a particular type of matrices is studied. It has
been analyzed from the point of view of its characterizations,
through the establishment of relations with different known types of
complex matrices, and this particular class of matrices has been
effectively obtained by different numerical methods. Now we detail
the specific problems developed in this thesis and the results
achieved.
First, a new class of matrices called {K,s+1}-potent has
been introduced. It can be seen that the {K,s+1}-potent
matrices contain, as particular cases, {s+1}-potent matrices,
regular matrices, centrosymmetric, mirrorsymetric, circulant, etc.
These latter types of matrices are very useful in various areas such
as multiconductor transmission lines, antennas, waves, electrical
and mechanical systems, and communication theory, etc.
For this class of {K,s+1}-potent matrices, it has been
performed an analysis about its existence. Also, different
properties related to the sum, product, conversely, conjugate
transpose, similarity, and direct sum, have been obtained.
The results that allow the further development of this thesis are
given by the characterizations of the {K,s+1}-potent
matrices. These necessary and sufficient conditions have been
obtained from different points of view: using spectral theory,
powers of matrices, generalized inverses, and by a block
representation of a matrix of index 1. This allow us to study the
particular cases previously mentioned.
Then, the new introduced class of matrices has been related to
different classes of matrices with complex coefficients:
{K}-Hermitian matrices,
{s+1}-generalized projectors,
unitary matrices, normal matrices, {K}-centrosymmetric
generalized matrices, etc. We have obtained a series of inclusions
between the sets that define them, and it has been observed that
most of these inclusions are strict constructing the appropriated
counterexamples to check them.
With the purpose to build in an effective way matrices of this
class, several algorithms have been developed for the case s
greater than or equals 1, and for the case s = 0.
Their effectiveness and performance have been analyzed through its
implementation in MATLAB.
Moreover, for the cases s greater than or equals 1 and s = 0
it has been solved the inverse problem for calculating involutive
matrices K that satisfy the matrix equation that we are
analyzing. These algorithms have been constructed from spectral
theory, particularly through the principal idempotents of the
original matrix.
This thesis ends with an extension of the previous study to the case
of {K,-(s+1)}-potent matrices, completing all possible
integer values of s. As before, in the last analysis, we have
distinguished the case s = 0 and s greater than or equals 1.
This thesis is organized in 5 chapters. Chapter 1 contains an
introduction where the background are mentioned, and the necessary
notation is introduced. Chapter 2 contains the existence and
properties of {K,s+1}-potent matrices. The main result in
this chapter, Theorem 2.2, provides characterizations of this class
of matrices. Chapter 3 introduces a number of sets of matrices and
establishes relations with the {K,s+1}-potent matrices by
using the characterizations obtained in Chapter 2. In Chapter 4
different algorithms for calculating the {K,s+1}-potent
matrices are developed. First, matrices of this class are
constructed from spectral information given by the involutive matrix
K. More examples can be constructed using: this algorithm, an
auxiliary algorithm that allows us to find another {K,s+1}-potent matrix,
and the analysis of linear combinations. In the
last part of Chapter 4, the inverse problem is analyzed and several
numerical examples are shown. Chapter 5 extends the type of matrices
introduced to include all cases for integer values of s. The
thesis concludes with an annex where the final conclusions and
future lines are shown.