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Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed. [ABSTRACT FROM AUTHOR]

The notions of C-tensor, C0-tensor and C ¯ -tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and C ¯ -tensor are provided. We next show that the sum of two C-tensors (C0-tensors) is a C-tensor (C0-tensor) while the Hadamard product of two C-tensors (C0-tensors) is not a C-tensor (C0-tensor). We also present a result that illustrates the Hadamard product of two C-tensor is again a C-tensor under some sufficient conditions. As an application of these classes of tensors, an exclusion interval for the real eigenvalues of a real tensor is proposed. Finally, we provide a necessary and sufficient condition for the exclusion interval to be nonempty. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H. Let B A (H) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several A -numerical radius inequalities for operator matrices in B A (H ⨁ H) . In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption 'A is strictly positive'. We then establish several new upper and lower bounds for the A -numerical radius of an operator matrix whose entries are operators in B A (H). Further, we prove some refinements of earlier A-numerical radius inequalities for operators in B A (H) . [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result. [ABSTRACT FROM AUTHOR]

The notion of the Moore–Penrose inverse of an even-order tensor and the two-term reverse-order law for the Moore–Penrose inverse of even-order tensors via the Einstein product were introduced, very recently. In this article, the Moore–Penrose inverse of an arbitrary tensor is introduced first. Various new expressions of the Moore–Penrose inverse are also proposed. A new generalized inverse of a tensor called product Moore–Penrose inverse is then introduced by extending the Moore–Penrose inverse of an arbitrary tensor. A necessary and sufficient condition for the coincidence of the Moore–Penrose inverse and the product Moore–Penrose inverse of an arbitrary tensor is also provided. Prior to these, a set of new sufficient conditions for computing the Moore–Penrose inverse of the product of two tensors via the Einstein product is illustrated. Finally, some necessary and sufficient conditions are obtained for the three-term reverse-order law of tensors via the same product. [ABSTRACT FROM AUTHOR]

In this article, we obtain some upper bounds for A-numerical radius of operators in semi-Hilbertian spaces which improve the existing ones. Furthermore, we discuss some generalizations of earlier A-numerical radius inequalities. We also establish several inequalities for B-numerical radius of n × n operator matrices, where B is an n × n diagonal matrix whose diagonal entries are positive operator A. [ABSTRACT FROM AUTHOR]

Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product. [ABSTRACT FROM AUTHOR]

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained. [ABSTRACT FROM AUTHOR]

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory. [ABSTRACT FROM AUTHOR]

In this article, a convergence theorem and comparison theorems for Brou-splittings and Bran-splittings (introduced in [D. Mishra, K.C. Sivakumar, Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings, Linear Algebra Appl. (accepted)]) are presented. We then conclude with a new characterization of range monotone matrices. [ABSTRACT FROM AUTHOR]

A dominance rule for singular matrices using proper splittings is proposed. This extends the corresponding notion, known for nonsingular matrices. An application to the nonnegativity of the Moore–Penrose inverse is presented. [ABSTRACT FROM AUTHOR]