For a given p×q complex matrix C, a necessary and sufficient condition is obtained for the existence of a matrix X satisfying JpX−XJq=C, here Jr denotes the r×r Jordan block of 0. An easy ...construction of the solution X is given if it exists. These results lead to a proof of the fact that a nilpotent matrix is similar to a direct sum of Jordan blocks.
In this paper, fuzzy linear singular differential equations under so-called granular differentiability are investigated in which the coefficients and initial conditions are as fuzzy numbers. Some new ...notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix are introduced. Moreover, we compare the proposed method with other ones based on other derivatives, using some examples.
It is well known that in CX, the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation ...between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications.
The admissible map that determines a map between classifying spaces of connected compact Lie groups can be regarded as a matrix. We discuss the diagonalizability of such matrices as well as the ...triangularizability, particularly in the case of generalized cyclic matrices. Sometimes a high–dimensional behavior characterizes the induced homomorphism of the cohomology. We will generalize some results in 8.
We analyse the spectral properties of solutions of the Yang-Baxter-like matrix
equation. We explore the solution set when A is nonsingular, give partial
results for nilpotent matrices, and construct ...elementary solutions to the
problem.
The synchronization of fractional-order complex networks with general linear dynamics under directed connected topology is investigated. The synchronization problem is converted to an equivalent ...simultaneous stability problem of corresponding independent subsystems by use of a pseudo-state transformation technique and real Jordan canonical form of matrix. Sufficient conditions in terms of linear matrix inequalities for synchronization are established according to stability theory of fractional-order differential equations. In a certain range of fractional order, the effects of the fractional order on synchronization is clearly revealed. Conclusions obtained in this paper generalize the existing results. Three numerical examples are provided to illustrate the validity of proposed conclusions.
•The synchronization of fractional-order complex networks is investigated.•Real Jordan canonical form of matrix is employed to convert error system.•Sufficient conditions in terms of linear matrix inequalities are established.•The effect of the derivative order on synchronization is clearly revealed.•The conditions adopted in this paper improve some existing results.
In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. If a∈R is a nilpotent element of index n with von Neumann regular an−1, we ...decompose a=ea+(1−e)a with ea∈eRe≅Mn(S) a Jordan block of size n over a corner S of R, and (1−e)a nilpotent of index <n for an idempotent e of R commuting with a. This result makes it possible to characterize prime rings of bounded index n with a nilpotent element a∈R of index n and von Neumann regular an−1 as a matrix ring over a unital domain.
Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict ...equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of m×n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure.
McDonald and Paparella Linear Algebra Appl. 498 (2016), 145-159 gave a necessary condition on the structure of the Jordan chains of h-cyclic matrices. In this work, that necessary condition is shown ...to be sufficient. As a consequence, we provide a spectral characterization of nonsingular, h-cyclic matrices.
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