Schur Forms and Normal-Nilpotent Decompositions

Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.

Linear Operators That Strongly Preserve Nilpotent Matrices over Antinegative Semirings

Let S be an antinegative commutative semiring without zero divisors and Mn（S） be the semiring of all n × n matrices over S. For a linear operator L on Mn（S）, we say that L strongly preserves nilpotent matrices in Mn（S） if for any A ∈ Mn（S）, A is nilpotent if and only if L（A） is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

On Fine Matrix Representations of Nilpotent Operators

In this paper we introduce the concepts of "O-state" and "I--state" for nilpotent operators and give a fine matrix representation for each nilpotent operator according to its state.Let X’, Y be infinite dimensional Banach spaces over complex field A. Denote by Y} the set of all bounded linear operators from X to Y. If X=Y, we write instead of B(X, F). For T^B(X"), let -D(T), ker T, R(T} denote the domain, kerner and the range of T respectively. For a subset MdX, M denotes the closure of M. Let H be a Hilbert space and MdH, then ML denotes the annihilator of If.

Subalgebras of Nilpotent Matrices

In this note we describe explicitly the subalgebras of nilpotent matrices and obtain some interesting results.

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpotent matrix N.

Some Polynomial Maps with Jacobian Rank Two or Three

We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure of polynomial maps H to H=(H_(1)(x_(1),x_(2),…,x_(n)),b_(3)x_(3)+…+b_(n)x_(n)+H^((0))_(2)(x_(2)),H_(3)(x_(1),x_(2)),…,H_(n)(x_(1),x_(2))).