Polynomial matrix - Wikipedia
In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices. A univariate polynomial matrix P of degree p is defined as:
The polynomial representation is enough to establish that circulant matri-ces commute in multiplication and that their product is also a polynomial in K T. That is to say (16) If X = x(K T) and Y = y(K T) are circulant matrices, then XY = YXis also a circulant matrix. The matrix operator K T has a spectral factorisation that is particularly
线性变换的矩阵表示 - 知乎 - 知乎专栏
在下文中， V 是一个定义在任意field F 上的dimension为 n 的linear space，而 T 为 V\rightarrow V 的linear transformation。 在介绍矩阵表示之前，我们首先要了解两个概念：minimal polynomial以及elementary divisors。 对于实数的多项式，我们并不陌生。 而对于linear transformatio. 在之前的一篇文章中，我已经提到过，linear transformation与矩阵是存在着对应关系的。 对于固定的basis，我们可以将每一个linear transformation写成一个矩阵。 当basis发生变化时，该linear …
linear algebra - matrix representation of polynomial
A matrix couples two vectors, outputting a degree-two polynomial. How to transform that matrix so it acts on a projection of those vectors?
A polynomial matrix is a polynomial with matrix-valued coefficients, e.g.: A(z) = 1 −1 −1 2 + 1 1 1 −1 z−1 + −1 2 1 −1 z−2; (1) a polynomial matrix can equivalently be understood a matrix with polynomial entries, i.e. A(z) = 1+z−1 −z−2 −1+z−1 +2z−2 −1+z−1 +z−2 2−z−1 −z−2 ; (2)
matrix polynomial and strictly matrix concave on D; (iii) G(x) is a matrix rational function and q-module matrix concave on D. Explicit constructions of semideflnite representations are given. Some examples are illustrated. Key words: convex sets, linear matrix inequality, matrix concavity, polynomial matrix inequality, semideflnite
1. Definition of polynomial representations Let F be an in nite eld. For each pair (i;j) with 1 i;j n, let X ij: GL n(F) !F be the coordinate function sending a matrix xto its entry x ij. Let F[X ij] denote the algebra generated by these functions. As this notation suggests, we identify F[X ij] with the polynomial ring in n2
Representations of polynomials. First, consider the different representations of polynomials, and the time necessary to complete operations based on the representation. There are 3 main representations to consider. 1. Coefficient vector with a monomial basis 2. Roots and a scale term • A (x)=(x−r. 0) · (x−r. 1) ·····x (−r. n− ...
Polynomial Matrix - an overview | ScienceDirect Topics
Let F be a field, F [λ] the algebra of polynomials in one variable λ with coefficients in F. Matrices with entries in F [λ] are called matrix polynomials, or polynomial matrices. We start with the Smith canonical form which plays an important role in the analysis of matrix polynomials.
Polynomial Matrix - an overview | ScienceDirect Topics
The polynomial matrix fraction description is a mathematically efficient and useful representation for a matrix of rational functions. With application to the transfer function of a multiinput, multioutput linear state equation, the polynomial fraction description can inherit the structural features that, for example, permit natural ...