Some norm inequalities concerning generalized inverses, 2

We extend to the von Neumann–Schatten classes C_p and norms ·_p, where 2  p < ∞, Penrose’s result on minimizing AXB − C₂. We give an example to show that this extension does not hold for 1  p < 2. The proof of the global inequality depends on local considerations.     

Recursive Formulas for the Generalized LM-Inverse of a Matrix

In this paper, we present recursive formulas for the sequential determination of the generalized LM-inverse of a general matrix. The formulas are developed for a matrix augmented by a column. These formulas are particularized to obtain also recursive relations for the generalized L-inverse of a general matrix augmented by a column.     

Least squares solutions with special structure to the linear matrix equation AXB = C

In this article we give some formulas for the maximal and minimal ranks of the submatrices in a least squares solution X to AXB = C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively. Finally, some Hermitian properties for least squares solution to matrix equation AXB = C are derived.     

Asymptotic properties of some subset vector autoregressive process estimators

We establish consistency and derive asymptotic distributions for estimators of the coefficients of a subset vector autoregressive (SVAR) process. Using a martingale central limit theorem, we first derive the asymptotic distribution of the subset least squares (LS) estimators. Exploiting the similarity of closed form expressions for the LS and Yule–Walker (YW) estimators, we extend the asymptotics to the latter. Using the fact that the subset Yule–Walker and recently proposed Burg estimators satisfy closely related recursive algorithms, we then extend the asymptotic results to the Burg estimators. All estimators are shown to have the same limiting distribution.     

Perturbation analysis for the generalized Schur complement of a positive semi‐definite matrix

Let and S=C?B^HA^?B be the generalized Schur complement of A?0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR‐95‐38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Fast enclosure for the minimum norm least squares solution of the matrix equation AXB = C

Fast algorithms for enclosing the minimum norm least squares solution of the matrix equation AXB = C are proposed. To develop these algorithms, theories for obtaining error bounds of numerical solutions are established. The error bounds obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. Copyright © 2015 John Wiley & Sons, Ltd.     

Recursive Determination of the Generalized Moore–Penrose <Emphasis Type="Italic">M</Emphasis>-Inverse of a Matrix

In this paper, we obtain recursive relations for the determination of the generalized Moore–Penrose M-inverse of a matrix. We develop separate relations for situations when a rectangular matrix is augmented by a row vector and when such a matrix is augmented by a column vector.     

Generalized intersection bodies

We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann–Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized Busemann–Petty problem. We show that these classes share many identical structural properties, proving the same results using integral geometry techniques for Zhang's class and Fourier transform techniques for Koldobsky's class. Using a functional analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the integral geometry of the Grassmann manifold.     

Co-Hopfian Modules of Generalized Inverse Polynomials

Let R be an associative ring not necessarily possessing an identity and (S, ≤) a strictly totally ordered monoid which is also artinian and satisfies that 0 ≤s for any s∈S. Assume that M is a left R-module having propertiy (F). It is shown that M is a co-Hopfian left R-module if and only if [M ^{S , ≤}] is a co-Hopfian left [[R ^{S , ≤}]]-module. Received October 14, 1998, Accepted October 15, 1999     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

Least‐squares solutions and least‐rank solutions of the matrix equation AXA* = B and their relations

A Hermitian matrix X is called a least‐squares solution of the inconsistent matrix equation AXA^* = B, where B is Hermitian. A^* denotes the conjugate transpose of A if it minimizes the F‐norm of B ? AXA^*; it is called a least‐rank solution of AXA^* = B if it minimizes the rank of B ? AXA^*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

Cholesky decomposition of a positive semidefinite matrix with known kernel

The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments.