Related papers: A Journey into Matrix Analysis
Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and…
This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…
Convenient and simple numerical techniques for performing quantum computations based on matrix representations of Hilbert space operators are presented and illustrated by various examples. The applications include the calculations of…
We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to…
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities…
In this paper, we present several new bounds for the norm and numerical radius of sums of Hilbert space operators. The obtained bounds form a new collection that enriches our understanding of these bounds. We compare our bounds with the…
Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal…
Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations…
The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schr\"odinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman…
The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its…
We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…
The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including…
Generalizing Pisier's idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal…
This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These…
Criteria and constructive methods for the completion of an incomplete basis of, or context in, a four-dimensional Hilbert space by (in)decomposable vectors are given.
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…
This work was originally published by the author in 1999 in a book [1] and later became part of the author's doctoral thesis in 1999 [2]. Since the original language of these works is not English, the author provides a translation of the…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…