Efficient analysis of structures holding tridiagonal and block tri-diagonal stiffness matrices, generalizing the method to other structures using householder and block householder transformation

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.