The inverses of Toeplitz band matrices

The elements of the inverse of a Toeplitz band matrix are given in terms ofthe solution of a difference equation. The expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix. Thus, the formulae are particularly simple for lower triangular and lower Hessenberg Toeplitz matrices. When the number of nonzero superdiagonals is small, sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive. If the inverse is positive, the row sums can be expressed in terms of the solution of the difference equation.     

Exact inverses of certain band matrices

Summary A method for determining exact inverses for arbitrary size band matrices of Toeplitz type and closely related types is outlined. A number of examples arising in statistical problems and finite differences are considered and the initial required elements of the inverses are given.     

Properties of the inverses of a set of band matrices

Those regions where the elements of the inverse of a Toeplitz matrix of band width four and order nalternate in sign are determined. A similar result concerning the elements of the inverse of a commonly occurring symmetric positive definite Toeplitz matrix of band width five and order nis extended to cover the case when the first and last rows of the matrix are modified through a change in the boundary conditions of the associated application. A method for economically obtaining the infinity norm of the pent-diagonal symmetric positive definite Toeplitz matrix is then derived.     

Extensions de demi-groupes inverses

We study the general problem of extension of one inverse semigroup by an another inverse semigroup. Any inverse semigroup can be rebuilt from its quotient by any congruence.     

Band matrices with Toeplitz inverses

It is shown that a square band matrix H=(h_ij) with h_ij=0 for j? i>r and i?j>s, where r+s is less than the order of the matrix, has a Toeplitz inverse if and only if it has a special structure characterized by two polynomials of degrees r and s, respectively.     

Toeplitz matrices with banded inverses

Given a Toeplitz matrix T with banded inverse [i.e., (T^?1)_ij=0 for j?i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient condition for the decay of the elements of T as one moves away from the diagonal.     

Toeplitz matrices with Toeplitz inverses revisited

A necessary and sufficient condition derived by Huang and Cline for a nonsingular Toeplitz matrix to have a Toeplitz inverse is shown to hold under more general hypotheses than indicated by them.     

Toeplitz matrices with totally nonnegative inverses

It is shown that the inverse of a Toeplitz matrix has only nonnegative minors if the zeros of a certain polynomial are positive or if their arguments are less than π?(k+n), where n is the dimension and k+1 is the bandwidth of the matrix.     

Extensions of matrix valued functions with rational polynomial inverses

Let R_j : |j| m, be a given set of n×n matrices. Necessary and sufficient conditions for the existence and uniqueness of an invertible function F() = F_j^j in the Wiener algebra of n×n matrix valued functions on the unit circle || = 1 such that F_j=R_j for |j| m, and F admits either a right or a left canonical factorization and the matrix Fourier coefficients of F^–1 vanish for |j| > m are presented and discussed. In the special case that the block Toeplitz matrix based on the given R_j is positive definite there is exactly one such extension: the so-called maximum entropy or autoregressive extension of statistical estimation theory. Some special properties of this extension are discussed.     

On the inverses of a class of toeplitz matrices of band width five

For a symmetric positive definite Toeplitz matrix of band width five and order n, those regions where the elements of the inverse alternative in sign are determined.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

Generalized inverses of stochastic matrices

This paper describes all stochastic matrices which have a stochastic semi-inverse and gives a method of constructing all such inverses. Then all stochastic matrices which have a stochastic Moore-Penrose inverse are described.     

On inverses of Hessenberg matrices

The lower half of the inverse of a lower Hessenberg matrix is shown to have a simple structure. The result is applied to find an algorithm for finding the inverse of a tridiagonal matrix. With minor modifications, the technique applies to block Hessenberg matrices.     

A new class of matrices with positive inverses

It is well known that irreducibly diagonally dominant matrices with positive diagonal and non-positive off-diagonal elements have positive inverses. A whole class of symmetric circulant and symmetric quindiagonal Toeplitz matrices with positive inverses which do not satisfy the above conditions is found.     

Generalized inverses of substochastic matrices

This paper looks at the question of when a substochastic matrix has a substochastic generalized inverse. This question is answered for several generalized inverses, including semiinverses, the Moore–Penrose inverse, and the group inverse. Methods for constructing all such inverses are given.     

Determinantal formulae for matrices with sparse inverses

The determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be “read off” from the graph of the inverse of the matrix. The only information used is the zero pattern of the inverse, and each zero pattern yields one or more corresponding formulae for the determinant.     

Treediagonal matrices and their inverses

A generalization of tridiagonal matrices is considered, namely treediagonal matrices, which have nonzero off-diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements. Some properties of treediagonal matrices are given, and their inverses are characterized and shown to have an interesting structure.     

Group inverses of certain nonnegative matrices

Nonnegative matrices with the property that the group inverse of the matrixis equal to a power of the matrix are characterized. The special case of the Moore- Penrose inverse is considered. The situation when the matrix is stochastic is examined.