Matrix polynomial information

In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial P ( x ) = ∑ i = 0 n...

polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is...

linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as...

identity matrix. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial...

In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots...

equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which...

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable...

theorem for polynomials. In statistics, the equation V a = y {\displaystyle Va=y} means that the Vandermonde matrix is the design matrix of polynomial regression...

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version...

called Hurwitz matrix corresponding to the polynomial p {\displaystyle p} . It was established by Adolf Hurwitz in 1895 that a real polynomial with a 0 >...

x^{n}-1)} . Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix P {\displaystyle P} : C = c 0 I...

In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for...

Cayley–Hamilton theorem the matrix exponential is expressible as a polynomial of order n−1. If P and Qt are nonzero polynomials in one variable, such that...

unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular...

n × n matrix has entries over a field with at least 2n + 1 elements (e.g. a 5 × 5 matrix over the integers modulo 11). det(A+t I) is a polynomial in t...

polynomials, of the matrix (with polynomial entries) XIn − A (the same one whose determinant defines the characteristic polynomial). Note that this Smith...

blocks of the Sylvester matrix is empty). There is no common convention for the discriminant of a constant polynomial (i.e., polynomial of degree 0). For small...

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes...

its characteristic polynomial has n {\displaystyle n} distinct roots in F {\displaystyle F} . Let A {\displaystyle A} be a matrix over F {\displaystyle...

In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that...

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a...

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal...

the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible...

side of equation (3) is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend...

efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional...