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In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that every square matrix over an infinite field is always representable as a sum of a diagonal...

In this paper, we show that a linear, bijective map f:Mn(C)→Mn(C) such that f(X)f(Y)=N whenever XY = M, for M, N rank-one nilpotents, is of the form f(X)=cU−1XU for an invertible U∈Mn(C) and a non-...

We obtain the formula for the inverse of polynomial maps of of the form where M(x) is a homogeneous of degree m nilpotent matrix, all of whose powers are exact; and we obtain recursion relations for the scalars cmk . Such maps have recently been considered by Connell and Zweibel, but our derivations is based on our earlier result that F −1(a) where x(t z a) is the unique solution, with initial condition x(0z a)=z, of the Wazewski differential equation dx/dt=Ft (x)−1 a=a−M(x)a, with vector parameter a. Basic to our method is the multilinear matrix function B(x y,…z) uniquely determined by M(x). We give a new proof that all powers of M are exact provided only that both M and M 2 are exact.

In this paper, we introduce the notion of the EP-nilpotent decomposition and present some of its applications. By applying the decomposition, we introduce two partial orders (the E-N partial order ...

Let A be a nilpotent matrix of index three and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a simplified matrix equation of the same type with A replaced by its Jordan for...

In this paper, we introduce the notion of a generalized infinite Jordan matrix which is an infinite analogon of a finite Jordan matrix. The main result states that for every upper triangular infinite matrix a, there exists a column finite matrix x such that is a generalized infinite Jordan matrix.

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

Let be a commutative ring with identity and let be quasi-order relation on . Then will denote the structural matrix algebra over . In this study, we show that, Theorem 4.1 of Benkovic [Jordan homomorphisms on triangular matrices. Linear and Multilinear Algebra. 2005;53:345–356] and Theorem 7 of Jacobson N. and Rickart C. [Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 1950;69:479–502] can be extended to two special cases, one for each theorem of structural matrix algebras.

Let be an -by- partial isometry whose numerical range is a circular disc with centre and radius . In this paper, we are concerned with the possible values of and . We show that must be if is at most and conjecture that the same is true for the general . As for the radius, we show that if , then the set of all possible values of is . Indeed, it is shown more precisely that for , , the possible values of are those in the interval . In the proof process, we also characterize -by- partial isometries which are (unitarily) irreducible. The paper is concluded with a question on the rotational invariance of nilpotent partial isometries with circular numerical ranges centred at the origin.

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G n , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.

A ray pattern A of order n is said to be spectrally arbitrary if given any monic nth degree polynomial f(x) with coefficients from ℂ, there exists a matrix realization of A such that its characteristic polynomial is f(x). An n × n ray pattern A is said to be minimally spectrally arbitrary if replacing any nonzero entry of A by zero destroys this property. In this article, several families of ray patterns are presented and proved to be minimally spectrally arbitrary. We also show that for n ≥ 5, when A n is spectrally arbitrary, then it is minimally spectrally arbitrary.

In this paper, we introduce a new class of upper triangular nilpotent matrices, etermine the eigenvalues of the Hermitian part of rotating the type of nilpotent matrices and relate the numerical range of that typical type of matrices to the orbit of a point mass under a central force.

We describe the sub-bimodules of matrix bimodules over two structural matrix rings. Structural matrix bimodules arise as particular such sub-bimodules, and we discuss when such a bimodule is faithful or indecomposable. As an application, we obtain a large class of rings whose Jordan isomorphisms are either ring isomorphisms or ring anti-isomorphisms. Complete upper block triangular matrix rings over 2-torsion-free indecomposable rings are elements of this class.

We show that for any two n × n square-zero matrices A and B over a division ring, if the right column spaces of AB and BA are the same, then the rank of AB is at most n/4, and if, in addition, the right null spaces of AB and BA are the same, then the rank of A + B is at most n/2. This generalizes some known results.

We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace-down method in a way similar to that in nonnegative linear algebra [G.F. Frobenius, Uber Matrizen aus nicht negativen Elementen. Sitzungsber. Kon. Preuss. Akad. Wiss., 1912, in Ges. Abh., Vol. 3, Springer, 1968, pp. 546–557; D. Hershkowitz and H. Schneider, Solutions of Z-matrix equations, Linear Algebra Appl. 106 (1988), pp. 25–38; H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey, Linear Algebra Appl. 84 (1986), pp. 161–189], characterizing the solvability in terms of supports and access relations. We give a description of the solution set as combination of the least solution and the eigenspace of the matrix, and provide a general algebraic setting in which this result holds.

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$ with nilpotent orbits in the case the $n \times n$ matrix $B$ has rank $n-2$. Moreover, we give some results on the maximal nilpotent orbit that $\nb$ intersects nontrivially., Comment: 14 pages

In this article, the nilpotent orbits of the Witt algebra W 1 are determined under the automorphism group over an algebraically closed field F of characteristic p > 3. In contrast with a finite num...

At first glance, this problem appears to be quite difficult. Beyond the likely difficulty of finding such a matrix N(x), it is not even immediately clear how one would prove that a matrix N(x) is actually a solution without a great deal of matrix algebra. How ever, this problem is not hard as it seems. In fact, it is one of a large class of problems that can be solved via a surprising method based upon single-variable calculus. In this Journal, Khan [2] used nilpotent matrices and Taylor series to find matrix functions satisfying the exponential functional equation, f(x + y) = f(x) f(y). His method is an example of a much more general theory of matrix power series due to Weyr [4], which can be used to find matrix functions satisfying a variety of functional equations. (Rinehart [3] gives an excellent survey of Weyr's approach. Higham [1, ch. 4] gives a more comprehensive account, as well as further applications.) We say that a set of real-valued functions {f(x)}ni=x C C??(R) satisfies an analytic functional equation E if there is an analytic function E such that

Let f be a noncommutative polynomial in two variables. Let be a multiplicative semigroup of linear operators on a finite-dimensional vector space and T a fixed linear operator such that f(T, S) is nilpotent for all S in . What can we say about the invariant subspace structure of We study special cases of this and other related conditions.

In earlier work, the possible spectra in a congruence class and the possible Jordan forms, among nilpotent matrices congruent to a given nilpotent matrix, were characterized. Here, we study the natural, but quite different, question of possible Jordan forms of matrices with a single nonzero eigenvalue (which may be taken to be 1) that are congruent to a given matrix with that eigenvalue. We focus upon a class of matrices that includes the essentially real matrices for which 0 is not in the field of values. Unlike the nilpotent case, the number of Jordan blocks, as well as their sizes, may change. Necessary conditions are derived and then it is shown that they are sufficient through a lengthy construction process. Some of the results go beyond the indicated case, but do not cover general matrices. §This work was done within the activities of Centro de Estruturas Lineares e Combinatorias da Universidade de Lisboa.

In 1975, Paul Halmos asked: when can a square matrix be written as a product of nilpotent matrices? This was answered indirectly by Sourour in 1992: if n ≥ 3 then any n × n singular matrix can be written in this way. In this article, we determine the best possible index for the nilpotents in such a product, and compare our work on linear transformations of a vector space with analogous results for transformations of a set.

We give a general geometrical procedure to construct nilpotent morphisms Φ : F → F(d), with F a vector bundle on ℙ1, obtaining an analog of the Jordan canonical form. We investigate the possible splitting types of F in dependence on the degeneration behavior of Φ. Applications to nilpotent matrices with an arbitrary number of variables are also given.

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A 1,A 2, an n × n nilpotent Toeplitz matrix Nn , and an n × n cyclic permutation matrix Sn (s) such that the numbers of flat portions on the boundaries of W(A 1⊕N n ) and W(A 2⊕S n (s)) are, respectively, 2(n − 2) and 2n. § This article was presented at the 8th Workshop on Numerical Ranges and Numerical Radii, July 15–17, 2006, Universitat Bremen, Germany.

For an n-by-n complex matrix A, we consider the numbers of line segments and elliptic arcs on the boundary @WðAÞ of its numerical range. We show that (a) if n � 4 and A has an (n � 1)-by-(n � 1) submatrix B with W(B) an elliptic disc, then there can be at most 2n � 2 line segments on @WðAÞ, and (b) if n � 3, then @WðAÞ contains at most (n � 2) arcs of any ellipse. Moreover, both upper bounds are sharp. For nilpotent matrices, we also obtain analogous results with sharper bounds.

We provide a detailed structural description of the nilpotent primitive subgroups of GL(n, 𝔽), where 𝔽 is a finite field.

The author regrets that the printed version of the above article contains some errors in the proof of Proposition 2.3 and would like to apologize for that. Moreover, to fill this gap at least a lit...

Index plays a fundamental role in the study of descriptor systems. For regular descriptor systems, calculation of the index can be performed by calculating the index of the nilpotent matrix obtained by means of the Weierstrass canonical form. Notwithstanding, if the system is not regular, there is no algebraic technique to determine the index of the system. A sufficient algebraic criterion is provided to determine the index of a general linear time-invariant descriptor systems. Thereafter, we provide an alternate but lucid proof of the fact that impulse controllability is necessary and sufficient for the existence of a semistate feedback such that the closed loop system is of the index at most one. Finally, a sufficient test for the existence of a semistate feedback such that the closed loop system is of the index at most two is provided. Examples are given to illustrate the presented theory.

In the paper we present an extension of Segal theorem on Lie algebra automorphisms of upper triangular nilpotent matrices to the whole class of nilpotent subalgebras generated by positive parts of indecomposable Chevalley algebras.

Let F be a field with at least k+2 elements and let A be an n×n matrix over F. Then rank A≤k if and only if for every invertible matrix B and every distinct nonzero α1,…αk+1∊F at least on of the matrices B+α1 A,…,B+αk+1 A is invertible. This extends the results on spectral characterization of matrices having rank 1 due to Deguang-Limaye [2] as well as the finite-dimensional versions of the results due to Jafarian-Sourour [5] and Limaye [6]. Our proof is much shorter. Using a similar idea we obtain also a characterization of rank one nilpotents among all nilpotents, thus simplifying the proof of the characterization of linear mappings preserving nilpotent matrices.

(1997). Classification of nilpotent linear spaces in M(4; C) Communications in Algebra: Vol. 25, No. 6, pp. 1919-1932.

By a theorem of Levitzki, we prove that a generalized strongly nilpotent matrix is similar to a upper triangular matrix with zero diagonal by a scalar matrix, which generalizes a recent result by van den Essen and Hubbers. We also show that a special case of a conjecture of Connell and Zweibel is true.

We prove that a polynomial map F = X + H ∈ k[X] with homogeneous H(k is a field of characteristic zero) is linear triangularizable if and only if the Jacobian matrix J(H) is a noncommutative-nilpotent matrix.

In this note we describe explicitly those linear spaces of nilpotent matrices, which are stable under conjugation by diagonal matrices. As an application, we deduce a generalized version of a result of Valcher and Fornasini on pairs of nonnegative nilpotent matrices.

We give necessary and sufficient conditions for a singular matrix over an arbitrary field to be factorized into a product of two nilpotent matrices with prescribed ranks.

Let M(n, C) be the vector space of n × n complex matrices and let G(r,s,t) be the set of all matrices in M(n, C) having r eigenvalues with positive real parts eigenvalues with negative real part and t eigenvalues with zero real part. In particularG(0,n,0) is the set of stable matrices. We investigate the set of linear operators on M(n, C) that map G(r,s,t) into itself. Such maps include, but are not always limited to similarities, transposition, and multiplication by a positive constant. The proof of our results depends on a characterization of nilpotent matrices in terms of matrices in a particular G(r,s,t), and an extension of a result about the existence of a matrix with prescribed eigenstructure and diagonal entries. Each of these results is of independent interest. Moreover, our char-acterization of nilpotent matrices is sufficiently general to allow us to determine the preservers of many other "inertia classes."

We show that, with the exception of 2 × 2 nonzero nilpotent matrices, every singular square matrix over an arbitrary field is a product of two nilpotent matrices.

(1990). Minimum Dimension for a Square Matrix of Order n. The College Mathematics Journal: Vol. 21, No. 1, pp. 28-34.

An algorithm is presented which enables the transformation of the state equations of linear multivariable systems to any of six canonical forms. It is based on transformation of a matrix to the Hermite normal form, and does not require matrix inversion.

In all previous work on Lyapunov exponents for stochastic systems, the results take the form of asymptotic power series expansions in the noise parameter. In this paper we consider a natural class ...

A characterization is given for all linear transformations on a matrix algebra which preserve (1) the trace (2) the unit matrix (3) the trace and the second elementary symmetric function of the eigenvalues.

In this note we show how the Frobenius inequality on rank function can be applied to determine the maximum cardinality of a setS of pairwise-commutative n×n nilpotent matrices with the property that no finite product of distinct elements of S equals zero.

A matrix of zero trace is, with certain exceptions, the commutator of nilpotent matrices. It may (so far as we know) be necessary to go outside the original field for the entries of the nilpotent matrices.

If A is an n×n matrix over a field F of positive characteristic p, then In =AB−BA, for some matrix B, iff p divides the size of each Jordan block of A.