Related papers: Proving the existence of eigenvalues and eigenvect…
In the recent paper \cite{1}, Denton et al. provided the eigenvector-eigenvalue identity for Hermitian matrices, and a survey was also given for such identity in the literature. The main aim of this paper is to present the identity related…
In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The…
By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion…
Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…
We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the…
The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.
We study the Weierstrass preparation and division theorems over arbitrary test rings and the local structure of singularities in the space of nondegenerate arcs on algebraic varieties. As an application, we prove a strengthened version of a…
We give a proof for the fundamental theorem of algebra,using the Fredholm index phenomena
We confirm rigorously the conjecture, based on numerical and asymptotic evidence, that all the eigenvalues of a certain non-self-adjoint operator are real.
Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described.
In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible…
The Eigenvalue Theorem shows that solving a zero-dimensional polynomial system can be recast as an eigenvalue problem. This paper explores the relation between the Eigenvalue Theorem and the work of Ludwig Stickelberger (1850-1936).
For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These…
There are several proofs of the Fundamental Theorem of Algebra, mainly using algebra, analysis and topology. In this article, we have shown that the Fundamental Theorem of Algebra can be proved using Nevanlinna's first fundamental theorem…
A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number…
The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic and homogenous. In particular we prove maximum and comparison principle, Holder…
This paper is concerned with {an extension and reinterpretation} of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. {We state} two general abstract results on…
In this paper, we prove a conjecture raised by Zhi-Wei Sun in 2018 by the eigenvectors-eigenvalues identity found by Denton, Parke, Tao and X. Zhang in 2019.
Two well known facts from elementary number theory are proven by using Bergman spaces.
We introduce the notion of weighted singular vectors and weighted uniform exponent with respect to a set of weights. We prove invariance of these exponents for affine subspaces and submanifolds inside those affine subspaces. For certain…