Three linear preservers on infinite triangular matrices

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability.     

Linear preservers of minimal rank

It is proved that a linear transformation on the vector space of upper triangular matrices that maps the set of matrices of minimal rank 1 into itself, and either has the analogous property with respect to matrices of full minimal rank, or is bijective, is a triangular equivalence, or a flip about the south-west north-east diagonal followed by a triangular equivalence. The result can be regarded as an analogue of Marcus–Moyls theorem in the context of triangular matrices.     

Term rank preservers of Boolean matrices

We obtain some characterizations of linear operators that preserve the term rank of Boolean matrices. That is, a linear operator over Boolean matrices preserves the term rank if and only if it preserves the term ranks 1 and k(≠1) if and only if it preserves the term ranks 2 and l(≠2). Other characterizations of term rank preservers are given.     

Biderivations on tensor products of algebras

Let A be a noncommutative unital prime algebra and let S be a commutative unital algebra over a field 𝔽. We describe the form of biderivations of the algebra A?S. As an application, we determine the form of commuting linear maps of A?S.     

Lattice tensor products. II

G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A?B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A?B, obtaining the isomorphism Id(A?B)≌A?Id B. This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A?B introduced in Part I of this paper.     

Linear operators that preserve maximal column ranks of nonnegative integer matrices

The maximal column rank of an by matrix over a semiring is the maximal number of the columns of which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.

     

A projection method to solve linear systems in tensor format

In this paper, we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions, but our method is not limited to this type of application. We present an iterative scheme, which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which—similarly to Krylov subspace methods—stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format, which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high‐dimensional convection–diffusion equations.Copyright © 2012 John Wiley & Sons, Ltd.     

Linear preservers of isomorphic types of lattices of invariant operator ranges

We describe all linear self-mappings of the space of bounded linear operators in an infinite dimensional separable complex Hilbert space which preserve the isomorphism class of the lattice of invariant operator ranges.     

Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an $\epsilon$-approximation for the $d$-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on $d$ and logarithmically on ${\epsilon }^{-1}$. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on $d$ and logarithmic dependence on ${\epsilon }^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $max(s,t)\ge 1$. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential $(s,t)$-weak tractability with $max(s,t)<1$ is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).     

A note on adjacency preservers on hermitian matrices over finite fields

It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form.