It is well-known how any symmetric matrix can be transformed into a similar tridiagonal one [1, 2]. This orthogonal similarity transformation forms the basic step for various algorithms. For example if one wants to compute the eigenvalues of a symmetric matrix, one can rst transform it into a similar tridiagonal one and then compute the eigenvalues of this tridiagonal matrix. Very recently an a...

 = αI +βT, where T is defined by the preceding formula. This matrix arises in many applications, such as n coupled harmonic oscillators and solving the Laplace equation numerically. Clearly M and T have the same eigenvectors and their respective eigenvalues are related by μ = α+βλ . Thus, to understand M it is sufficient to work with the simpler matrix T . Eigenvalues and Eigenvectors of T Usu...

The inverse C = [ci,j ] of an irreducible nonsingular symmetric tridiagonal matrix is a socalled Green’s matrix, i.e. it is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for i ≤ j. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case....

We give a matrix factorization for the solution of the linear system Ax = f , when coefficient matrix A is a dense symmetric positive definite matrix. We call this factorization as "WW T factorization". The algorithm for this factorization is given. Existence and backward error analysis of the method are given. The WDWT factorization is also presented. When the coefficient matrix is a symmetric...

In this paper we study the connection between matrix measures and random walks with a block tridiagonal transition matrix. We derive sufficient conditions such that the blocks of the n-step block tridiagonal transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by m...

Let A be a tridiagonal matrix of order n. We show that it is possible to compute and hence condo (A), in O(n) operations. Several algorithms which perform this task are given and their numerical properties are investigated. If A is also positive definite then I[A-[[o can be computed as the norm of the solution to a positive definite tridiagonal linear system whose coeffcient matrix is closely r...

The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal mat...

It is well known that the inverse C = c i;j ] of an irreducible nonsingular symmetric tridiagonal matrix is a Green matrix, i.e. it satisses c i;j = u i v j for i j and two sequences of real numbers fu i g and fv i g. A similar result holds for nonsymmetric matrices A. There the inverse is given by four sequences fu i g; fv i g; fy i g; and fy i g: Here we characterize certain properties of A i...