A large group of structures hold tri-diagonal stiffness MATRICES. The eigenpairs and inverse of these MATRICES are found simpler than the ones of common MATRICES. In addition, using the householder transformation, symmetric MATRICES can be converted to the similar tri-diagonal MATRICES. Therefore, since stiffness MATRICES are symmetric, they can be changed to the similar tri-diagonal ones. In other words, all symmetric MATRICES can be converted to the tridiagonal ones and the simpler solution of tri-diagonal MATRICES can be used for all stiffness MATRICES. Such a comparison is also true for block tri-diagonal MATRICES and block symmetric MATRICES. Although block MATRICES are a specific kind of common MATRICES, we want to study them independently because working with blocks can be more time-saving and efficient in many cases. In this paper, efficient solutions are presented for tri-diagonal and block tridiagonal MATRICES. Besides, using the features of symmetric and block symmetric MATRICES they are converted to the tri-diagonal and block tri-diagonal ones.

Let c0 be the real vector space of all real sequences which converge to zero. For every x, yÎc0, it is said that y is block diagonal majorized by x (written y<b x) if there exists a block diagonal row stochastic matrix R such that y=Rx. In this paper we find the possible structure of linear functions T: c0®c0 preserving <b.

Many structural models such as grids, barrel vaults, trusses and frames with repetitive units, known as regular structures, have structural MATRICES in the form of M = F(B; A; BT ). In this paper, a simple and efficient method is presented for calculating the eigenvalues of the adjacency and Laplacian MATRICES of regular structures. These eigenvalues can be used in studying the combinatorial properties of these structures. Examples are included to show the accuracy of the presented approach.

In this paper, we consider an inverse eigenvalue problem (IEP) for constructing a special kind of acyclic matri-ces. The problem involves the reconstruction of matri-ces whose graph is a banana tree. This is performed by using the minimal and maximal eigenvalues of all lead-ing principal subMATRICES of the required matrix. The necessary and su cient conditions for the solvability of the problem is derived. An algorithm to construct the solution is provided.

In this paper, we consider characterizations based on the Bhattacharyya MATRICES. We characterize, under certain constraint, distributions such as normal, compound poisson and gamma via the diagonality of the 2 x 2 Bhattacharyya matrix.