Related papers: Condition Numbers of Hessenberg Companion Matrices
In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a new condition, generalizing known…
In recent years there has been a growing interest in companion matrices. There is a deep knowledge of sparse companion matrices, in particular it is known that every sparse companion matrix can be transformed into a unit lower Hessenberg…
The Frobenius companion matrix, and more recently the Fiedler companion matrices, have been used to provide lower and upper bounds on the modulus of any root of a polynomial $p(x)$. In this paper we explore new bounds obtained from taking…
We demonstrate a new kind of companion matrix, for polynomials of the form $c(\lambda) = \lambda a(\lambda)b(\lambda) + c_0$ where upper Hessenberg companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This construction…
We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for…
In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…
We study a collection of Hessenberg varieties in the type A flag variety associated to a nonzero semisimple matrix whose conjugacy class has minimal dimension. We prove each such minimal semisimple Hessenberg variety is a union Richardson…
The research aims to construct a new type of matrix called the Fibonacci-Hessenberg-Lorentz matrix by multiplying Fibonacci-Hessenberg matrices with Lorentz matrix multiplication. The study will start by examining the properties of…
This article study the average conditioning for a random underdetermined polynomial system. The expected value of the moments of the condition number are compared to the moments of the condition number of random matrices. An expression for…
We consider a particular type of matrices which belong at the same time to the class of Hessenberg and Toeplitz matrices, and whose determinants are equal to the number of a type of compositions of natural numbers. We prove a formula in…
Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of…
Estimating the condition numbers of random structured matrices is a well known challenge, linked to the design of efficient randomized matrix algorithms. We deduce such estimates for Gaussian random Toeplitz and circulant matrices. The…
A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are…
We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical…
In the high-dimensional data setting, the sample covariance matrix is singular. In order to get a numerically stable and positive definite modification of the sample covariance matrix in the high-dimensional data setting, in this paper we…
It is well known that square matrices with independent and identically distributed (iid) random entries are typically well conditioned. A natural question is whether this favorable behavior persists for random matrices whose entries obey…
We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many…
Companion matrices of the second type are characterized by properties that involve bilinear maps.
In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schroder and Fine number sequences. Algebraic proofs of these results can be given which make use…
In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…