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.