On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

On the decomposable numerical range of λI-N

Let N be an nxn normal matrix. For 1≤m≤n we characterize the convexity of the mth decomposable numerical range of λI_n -N which is defined to be

{det(X?(λI_n ?N)X) [sdot] X?C _n×m X ^? X=I_m }.

A related problem on mixed decomposable numerical range of λI_n -N is also discussed.     

Geometry of 2 × 2 hermitian matrices

LetD be a division ring which possesses an involution a → α . Assume that is a proper subfield ofD and is contained in the center ofD. It is pointed out that ifD is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions over F and that if D is of characteristic two,D is a separable quadratic extension ofF. Thus the trace map Tr:D → F, a → a + a is always surjective, which is formerly posed as an assumption in the fundamental theorem of n×n hermitian matrices overD when n ≥ 3 and now can be deleted. WhenD is a field, the fundamental theorem of 2 × 2 hermitian matrices overD has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized quaternions of characteristic not two This research was completed during a visit to the Academy of Mathematics and System Sciences, Chinese Academy of Sciences.     

A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.     

Geometry of 2×2 Hermitian matrices II

Let Z be a field of characteristic ≠2, D be a quaternion division algebra over Z and have a nonstandard involution of the first kind. The fundamental theorem of geometry of 2× 2 Hermitian matrices over D are proved. Thus, if D is a quaternion division algebra over Z with an involution of the first kind, then the fundamental theorem of geometry of 2× 2 Hermitian matrices over D are obtained.     

A Bang-Bang representation for 3×3 embeddable stochastic matrices

Summary It is proved that a 3×3 embeddable stochastic matrix has a representation as a product of a finite number of elementary stochastic matrices, with only one off-diagonal element positive. In particular if the determinant is 1/2 then only 6 matrices are needed and a necessary and sufficient condition for embeddability in this case is given.     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

When is the numerical range of a nilpotent matrix circular?

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces tr M^∗M² and tr M^∗M³ are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5×5 nilpotent matrix has circular numerical range are thoroughly discussed.     

A study of the (Δ, C)-equivalence of certain numerical matrices

We study the properties of (,C)-transforms of numerical matrices. It is proved that the set of invertible -matrices for a fixed (x) forms a group. On the basis of the properties obtained it is proved that semiscalar equivalence of two nonsingular equivalent polynomial matrices with the same invariant factor is equivalent to the (, C)-equivalence of the matrices of values of the last rows of the left generators of these matrices in Smith form.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 25–29.     

Solutions of Indefinite Hammerstein Equations

Consider the following equation:For positive definite or quasi-positive definite kernel k(x,y),the theory of equa-tion(1)has much been developed.But up to now,little is known about the caseof indefinite kernels.In this paper we get several results of the latter type. Let G R~N have finite measure and f(x,u)satisfy Caratheodory condition,     

Criteria for unitary congruence of complex 2 × 2 matrices

Canonical forms of 2 × 2 matrices with respect to unitary congruence transformations are described. This makes it possible to formulate simple criteria for checking whether the given matrices are unitarily congruent.     

Some generators for the algebra of n×n matrices

In this note, we show how the algebra of n×n matrices over a field can be generated by a pair of matrices A B, where A is an arbitrary nonscalar matrix and B can be chosen so that there is the maximum degree of linear independence between the higher commutators of B with A.     

Computation of the commutator of 2 × 2 matrices via five multiplications

Algorithms for computing the commutator AB ? BA of 2 × 2 matrices A and B are proposed that involve five multiplications.     

Characterizations of perturbations of spectra of 2×2 upper triangular operator matrices

When $A\in B\left(H\right)$ and $B\in B\left(K\right)$ are given, we denote by $M_C$ the operator acting on the infinite dimensional separable Hilbert space $H\oplus K$ of the form $M_C=\left(\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right)$. In this paper, we first give some necessary and sufficient conditions for $M_C$ to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some $C\in B(K,H)$, which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples.     

Perturbation of spectra for a class of 2×2 operator matrices

In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M _X = ( _{0 B} ^{A X} ) and M _Z = ( _{Z B} ^{A C} ) on the Hilbert space H ?? K and the sets $\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} $ and $\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} $ , where R(C) is a closed subspace, are characterized