In this paper, we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse. For this purpose, considering usual continued fraction, we define backward continued fraction for a real number and give some basic results on backward continued fraction. We give the relationships between the usual and backward continued fractions. Then we reobtain the LU fact...

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy conditions (i), (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V wit...

The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by u...

The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by u...

In this paper, we consider matrices whose inverses are tridiagonal Z{matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z{matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type D matrix and a ipped weak type D matrix whose parameters satis...

In this paper we construct the symmetric quasi anti-bidiagonal matrix that its eigenvalues are given, and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues. Not only elements of the tridiagonal matrix come from quasi anti-bidiagonal matrix, but also the places of elements exchange based on some co...

A little known property of a pair of eigenvectors (column and row) of a real tridiagonal matrix is presented. With its help we can define necessary and sufficient conditions for the unique real tridiagonal matrix for which an approximate pair of complex eigenvectors are exact. Similarly we can designate the unique real tridiagonal matrix for which two approximate real eigenvectors, with differe...

In this paper, we consider matrices whose inverses are tridiagonal Z–matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z–matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type D matrix and a flipped weak type D matrix whose parameters sat...

We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate a...