China
Africa
Explore chapters and articles related to this topic
Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
Two real or complex matrices A1 and A2 are similar if and only if the polynomial matrices (sI — A1) and (sI — A2) have the same invariant polynomials [11]. The invariant polynomials of (sI — A) are also known as the invariant factors of A since they are factors of the characteristic polynomial det(sI — A): det(sI — A) = φ1(s)φ2(s) … φn(s). Also, φi+1(s) is a factor of φi(s), and φ1(s) is the minimal polynomial of A, the monic polynomial p(s) of least degree such that p(A) = 0.
Canonical Forms
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Facts requiring proof for which no specific reference is given can be found in [HK71, Chapter 7] or [DF04, Chapter 12]. Let M ∈ F[x]n×n. Then M has a unique Smith normal form.Let A ∈ Fn×n. There are no zeros on the diagonal of the Smith normal form of xIn – A.(Division Property) If a(x), b(x) ∈ F[x] and b(x) ≠ 0, then there exist polynomials q(x), r(x) such that a(x) = q(x)b(x) + r(x) and r(x) = 0 or deg r(x) < deg b(x).The Smith normal form of M = xI – A and, thus, the Smith invariant factors of A can be computed as follows: For k = 1, …, n – 1 – Use elementary row and column operations and the division property of F[x] to place the greatest common divisor of the entries of M[{k, …, n}] in the kth diagonal position.– Use elementary row and column operations to create zeros in all nondiagonal positions in row k and column k.Make the nth diagonal entry monic by multiplying the last column by a nonzero element of F.
Finite-time estimation for linear time-delay systems via homogeneous method
Published in International Journal of Control, 2019
This paper adopts the method based on ring theory since it enables us to reuse some useful techniques developed for linear systems without delays. The following notations will be used in this paper. is the field of real numbers. The set of positive integers is denoted by . Ip means p × p identity matrix. is the polynomial ring over the field and is the -module whose elements are vectors of dimension n and whose entries are polynomials. By we denote the set of matrices of dimension q × s, whose entries are in . For a matrix M(δ), means the rank of the matrix M(δ) over . We denote as the set of invariant factors of the Smith form of M(δ).