Related papers: On Determining the Eigenprojection and Components …
An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the ith column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur…
The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular…
By using the methods of Cauchy-Binet type formula and adjugate matrix respectively, a wonderful equality relating to the elements of eigenvectors, the eigenvalues and the submatrix eigenvalues is proved in arXiv:1908.03795. In the note, we…
The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and,…
We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for…
We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key…
When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, the principal $p$th root of $A$ can be computed by Schr\"oder's method, among many other methods. In this paper we present a further study of Schr\"oder's method for the…
Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l…
An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of…
A simple approximate relationship between the ground-state eigenvector and the sum of matrix elements in each row has been established for real symmetric matrices with non-positive off-diagonal elements. Specifically, the $i$-th components…
If a nonnegative selfadjoint linear relation $A$ in a Hilbert space and a closed subspace $\mathcal{S}$ are assumed to satisfy that the domain of $A$ is invariant under the orthogonal projector onto $\mathcal{S},$ then $A$ admits a…
We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with…
In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used…
We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ…
The Schur-Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive…
Let $A$ be a rectangular matrix of size $m\times n$ and $A_1$ be the random matrix where each entry of $A$ is multiplied by an independent $\{0,1\}$-Bernoulli random variable with parameter $1/2$. This paper is about when, how and why the…
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…
Conditions for existence and formulas for the first- and second order total derivatives of the eigenvalues, and the first order total derivatives of the eigenprojections of smooth matrix-valued functions $H\colon\Omega\to S(m)$ are given.…