The decompositions of soft sets and soft matrices are important tools for theoretical and practical studies. In this paper, firstly, we study the decomposition of soft sets in detail. Later, we introduce the concepts of $\alpha$-upper, $\alpha$-lower, $\alpha$-intersection and $\alpha$-union for soft matrices and present some decomposition theorems. Some of these operations are set-restricted types of existing operations of soft sets/matrices, others are $\alpha$-oriented operations that provide functionality in some cases. Moreover, some relations of decompositions of soft sets and soft matrices are investigated and the newfound relations are supported with numerical examples. Finally, two new group decision making algorithms based on soft sets/matrices are constructed, and then their efficiency and practicality are demonstrated by dealing with real life problems and comparison analysis. By using these proposed approaches, solutions can be presented to soft set-based multi-criteria decision making problems, both ordinary and involving primary assessments. These allow to handle soft set-based multi-criteria decision making from different perspectives.

In this work, we introduce some useful concepts  in soft set theory, such as $\overline{\alpha}$-inverse intersection and  $\overline{\alpha}$-inverse union of inverse soft sets, together with  the type soft $\overline{\alpha}$-upper, $\overline{\alpha}$-lower, $\overline{\alpha}$-intersection and $\overline{\alpha}$-union of inverse soft matrices.  Our main contribution is of proposing a new decision-making method associated with  inverse soft sets and inverse soft matrices.

In the present communication, we have introduced the notion of T-spherical fuzzy soft matrix (TSFSM) and studied various types of associated binary operations and properties. In literature, it has been observed that the concept of soft matrix plays a vital role in many engineering applications as well as to cater different socio-economic and financial sector problems. As per the definition of T-spherical fuzzy set, the proposed notion would have an additional capability to address the impreciseness of the information close enough to human opinion mathematically.Further, on the basis of the structure of proposed TSFSM and using the concept of choice matrix along with its weighted form, a new algorithm for the decision-making process has been outlined. Next, utilizing the score/utility matrix, we present another algorithm for the selection process. For the sake of understanding of the proposed methodologies, illustrative examples have also been presented. Some comparative remarks for the proposed techniques in contrast with existing techniques have been listed for a better readability and understanding.

In this paper, we focus on the matrices representing the inverse fuzzy soft sets over both the universal object set and the universal parameter set. Some basic operations and properties of these inverse fuzzy soft matrices were investigated. Moreover, two adjustable approaches to Multi-Criteria Group Decision Making (MCGDM), namely Inverse Fuzzy soft Sum-Product Decision Making (IFSSPDM) and Inverse Fuzzy soft Distributive If-di, erence Decision Making (IFSDIf-dDM), were developed. The IFSSPDM approach achieved the optimal choice for the MCGDM problem based on the inverse fuzzy soft structures consisting of multiple-discrete parameter sets and common universal object sets. The objective of IFSDIf-dDM approach was to present a solution to the MCGDM problem based on the inverse fuzzy soft structures consisting of a common universal parameter set and two discrete universal object sets. Thus, the solutions could be obtained using the practicality of inverse fuzzy soft matrices for two di, erent types of decision making problems. Besides, the comparisons made showed that the proposed approaches produced more convincing outputs than the current fuzzy soft approaches.

In this paper, a class of matrices, namely, Drazin-dagger matrices, in which the Drazin inverse andthe Moore-Penrose inverse commute, is introduced. Also, some properties of the generalized inverses of these matrices, are investigated. Moreover, some results about the Moore-Penrose inverse, the Drazin inverse and the numerical range of some reciprocal matrices are obtained. In particular, the relations between reciprocal matrices, Drazin-Dagger matrices and star order are established. Also, some properties of the generalized inverses of the conjugate EP matrices are studied. To illustrate the results, some numerical examples are also given.

Please click on PDF to view the abstract

The relations between the spectrum of the matrix Q+ Rand the spectra of matrices (g+δ) Q+(a+b) R-QR-RQ, QR-RQ, aRb-QRQ, aRQR-(QR) 2, and bR-QR have been given assuming that the matrix Q+R is diagonalizable, where Q, R are {g,δ}-quadratic matrix and f; g-quadratic matrix, respectively, of order n.