In this paper, we deal with Leslie and doubly Leslie matrices of order n. In particular, with the companion and doubly companion matrices. We study three inverse eigenvalues problems which consist of ...constructing these matrices from the maximal eigenvalues of its all leading principal submatrices. For Leslie and doubly companion matrices, an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered, and for the doubly Leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n−1 is required. We give necessary and sufficient conditions for the existence of a Leslie matrix and a companion matrix, and sufficient conditions for the existence of a doubly Leslie matrix and a doubly companion matrix. Our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices.
Let p and q be polynomials with degree 2 over an arbitrary field F, such that p(0)q(0)≠0. A square matrix with entries in F is called a (p,q)-product when it can be split into AB for some pair (A,B) ...of square matrices such that p(A)=0 and q(B)=0.
A (p,q)-product is called regular when none of its eigenvalues is the product of a root of p and of a root of q. A (p,q)-product is called exceptional when all its eigenvalues are products of a root of p and of a root of q. In a previous work 6, we have shown that the study of (p,q)-products can be entirely reduced to the one of regular (p,q)-products and to the one of exceptional (p,q)-products. Moreover, regular (p,q)-products have been characterized in 6 thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment.
The present article completes the study of (p,q)-products by obtaining a complete characterization of exceptional (p,q)-products. Beforehand, only very special cases in this problem had been solved, most notably the ones of products of two involutions, products of two unipotent elements of index 2, and products of a unipotent element of index 2 with an involution.
The study involves tools and strategy that are similar to the ones used for the exceptional (p,q)-sums undertaken in a recent article 7.
The Fiedler matrices of a monic polynomial p(z) of degree n are n×n matrices with characteristic polynomial equal to p(z) and whose nonzero entries are either 1 or minus the coefficients of p(z). ...Fiedler matrices include as particular cases the classical Frobenius companion forms of p(z). Frobenius companion matrices appear frequently in the literature on control and signal processing, but it is well known that they posses many properties that are undesirable numerically, which limit their use in applications. In particular, as n increases, Frobenius companion matrices are often nearly singular, i.e., their condition numbers for inversion are very large. Therefore, it is natural to investigate whether other Fiedler matrices are better conditioned than the Frobenius companion matrices or not. In this paper, we present explicit expressions for the condition numbers for inversion of all Fiedler matrices with respect the Frobenius norm, i.e., ‖A‖F=∑ij|aij|2. This allows us to get a very simple criterion for ordering all Fiedler matrices according to increasing condition numbers and to provide lower and upper bounds on the ratio of the condition numbers of any pair of Fiedler matrices. These results establish that if |p(0)|⩽1, then the Frobenius companion matrices have the largest condition number among all Fiedler matrices of p(z), and that if |p(0)|>1, then the Frobenius companion matrices have the smallest condition number. We also provide families of polynomials where the ratio of the condition numbers of pairs of Fiedler matrices can be arbitrarily large and prove that this can only happen when both Fiedler matrices are very ill-conditioned. We finally study some properties of the singular values of Fiedler matrices and determine how many of the singular values of a Fiedler matrix are equal to one.
Let p and q be polynomials with degree 2 over an arbitrary field F. A square matrix with entries in F is called a (p,q)-sum when it can be split into A+B for some pair (A,B) of square matrices such ...that p(A)=0 and q(B)=0.
A (p,q)-sum is called regular when none of its eigenvalues is the sum of a root of p and of a root of q. A (p,q)-sum is called exceptional when each one of its eigenvalues is the sum of a root of p and of a root of q. In a previous work 8, we have shown that the study of (p,q)-sums can be entirely reduced to the one of regular (p,q)-sums and to the one of exceptional (p,q)-sums. Moreover, regular (p,q)-sums have been characterized thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment.
The present article completes the study of (p,q)-sums by characterizing the exceptional ones. The new results here deal with the case where at least one of the polynomials p and q is irreducible over F.
Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in Hλ, and let A∈Hn×n be nonderogatory. In this paper we consider the problem of finding a solution X∈Hn×n to p(X)=A. A necessary condition ...for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well.
One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.
Algebraic linearizations of matrix polynomials Chan, Eunice Y.S.; Corless, Robert M.; Gonzalez-Vega, Laureano ...
Linear algebra and its applications,
02/2019, Letnik:
563
Journal Article
Recenzirano
Odprti dostop
We show how to construct linearizations of matrix polynomials za(z)d0+c0, a(z)b(z), a(z)+b(z) (when deg(b(z))<deg(a(z))), and za(z)d0b(z)+c0 from linearizations of the component parts, a(z) and b(z). ...This allows the extension to matrix polynomials of a new companion matrix construction.
Let Fq be the finite field with q elements (where q is a prime power). Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear ...group, GLn(Fq), on Gq(k,n) (Grassmann variety). The orbits of a subgroup of GLn(Fq) acting on the Grassmann variety are called (subspace) orbit codes. When the subgroup acting on Gq(k,n) is cyclic the associated codes are called cyclic orbit codes. We make a construction abelian non-cyclic orbit codes by making full use of the companion matrix of a primitive polynomial over finite fields, it is a partial spread code. Based on this code, an optimal partial spread code is obtained. Our results answer the first of two open problems presented by Climent et al.
In this paper, we present the concept of perturbation bounds for the right eigenvalues of a quaternionic matrix. In particular, a Bauer-Fike-type theorem for the right eigenvalues of a diagonalizable ...quaternionic matrix is derived. In addition, perturbations of a quaternionic matrix are discussed via a block-diagonal decomposition and the Jordan canonical form of a quaternionic matrix. The location of the standard right eigenvalues of a quaternionic matrix and a sufficient condition for the stability of a perturbed quaternionic matrix are given. As an application, perturbation bounds for the zeros of quaternionic polynomials are derived. Finally, we give numerical examples to illustrate our results. Key words. quaternionic matrices, left eigenvalues, right eigenvalues, quaternionic polynomials, Bauer-Fike theorem, quaternionic companion matrices, quaternionic matrix norms
The centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field. It is also known for the ...real Jordan canonical form. In this paper we characterize the centralizer of an endomorphism over an arbitrary field, and compute its dimension. The result is obtained via generalized Jordan canonical forms (for separable and nonseparable minimal polynomials). In addition, we also obtain the corresponding generalized Weyr canonical forms and the structure of its centralizers, which in turn allows us to compute the determinant of its elements.
Let
p
and
q
be polynomials with degree 2 over an arbitrary field
F
, and
M
be a square matrix over
F
. Thanks to the study of an algebra that is deeply connected to quaternion algebras, we give a ...necessary and sufficient condition for
M
to split into
A
+
B
for some pair (
A
,
B
) of square matrices over
F
such that
p
(
A
)
=
0
and
q
(
B
)
=
0
, provided that no eigenvalue of
M
splits into the sum of a root of
p
and a root of
q
. Provided that
p
(
0
)
q
(
0
)
≠
0
and no eigenvalue of
M
is the product of a root of
p
with a root of
q
, we also give a necessary and sufficient condition for
M
to split into
AB
for some pair (
A
,
B
) of square matrices over
F
such that
p
(
A
)
=
0
and
q
(
B
)
=
0
. In further articles, we will complete the study by lifting the assumptions on the eigenvalues of
M
.
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