A note on generalized companion pencils in the monomial basis
نویسندگان
چکیده
منابع مشابه
A note on companion pencils
Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding polynomial. In the classical companion pencil case (A,B) only the coefficient of the highest degree appears in B’s lower right corner. We will show, however, that all coefficients of the polynomial ...
متن کاملGeneralized Companion Matrices in the Lagrange Basis
It is well known that solving polynomial equations, or finding the eigenvalues of matrix polynomials, can be done by transforming to a generalized eigenvalue problem (see for example [10]). In this paper we examine a new way to do this directly from the values of the polynomial or matrix polynomial at distinct evaluation points.
متن کاملBlock LU factors of generalized companion matrix pencils
We present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems. In particular, we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange, and Bernstein/Bézier bases. Additionally, we give a pair of explicit LU factors associate...
متن کاملA Note on the Monomial Conjecture
Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed. Let (A,m, k) be a noetherian local ring of dimension n, m its maximal ideal and k = A/m. The Monomial Conjecture (henceforth MC) of Hochster asserts that, given any system of parameters (henceforth s.o.p.) x1, . . . , xn of A, (x1x2 . . . xn) t−1 / ∈ (x1, . . . , xn) ∀t > 0. Ho...
متن کاملA note on companion matrices
We show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices, n− 1 of them being the identity matrix in which a 2 × 2 identity submatrix in two consecutive rows (and columns) is replaced by an appropriate 2 × 2 matrix, the remaining being the identity matrix with the last entry replaced by possibly different entry. By a certain similarity tran...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
سال: 2019
ISSN: 1578-7303,1579-1505
DOI: 10.1007/s13398-019-00760-y