Pairs of mutually annihilating operators

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S^-1AS,S^-1BS).     

Determinant of a matrix that commutes with a Jordan matrix

Given a Jordan matrix J, we obtain an explicit formula for the determinant of any matrix T that commutes with it.     

Jordan structures of strictly lower triangular completions of nilpotent matrices

This work is part of a doctoral thesis, written under the supervision of Prof. A. Berman. It was supported by the Fund for Promotion of Research at the Technion.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Invariant subspaces of nilpotent operators and LR-sequences

The aim of this paper is to study systematically invariant subspaces of finitedimensional nilpotent operators. Our main motivation comes from classifying the similarity orbit in thelattice of invariant subspaces of a given nilpotent operator. We give a detailed study of the Littlewood-Richardson similarity orbit. We show that none of the natural similarity relations is equivalent with the others.     

Varieties of pairs of nilpotent matrices annihilating each other

We classify the irreducible components of the varieties \[ \V(n,a,b) = \{ (A,B) \in \M_n(\field) \times \M_n(\field) \mid AB = BA = A^a = B^b = 0 \}. \]     

An extremal sparsity property of the Jordan canonical form

We prove that among all the matrices that are similar to a given square complex matrix, the Jordan canonical form has the largest number of off-diagonal zero entries. We also characterize those matrices that attain this largest number.     

Spectral criteria for complementary triangular forms

In this paper we consider the following problem: Given two matricesA,Z∈? ^n×n , does there exist an invertiblen×n-matrixS such thatS ^?1 AS is an upper triangular matrix andS ^?1 ZS is a lower triangular matrix, and if so, what can be said about the order in which the eigenvalues ofA andZ appear on the diagonals of these triangular matrices? For special choices ofA andZ a complete solution is possible, as has been shown by several authors. Here we follow a lead, provided by Shmuel Friedland, who discussed the case where bothA andZ have at leastn-1 linearly independent eigenvectors, and we descibe the problem in terms of Jordan chains and left-Jordan chains for the matricesA, Z. The results give some insight in the question why certain classes of matrices (like the nonderogatory and the rank 1 matrices) allow for a detailed solution of the problems described above; for some of these classes the result of this analysis is presented here for the first time.     

Extended rank reduction formulas containing Wedderburn and Abaffy–Broyden–Spedicato rank reducing processes

The Wedderburn rank reduction formula and the Abaffy–Broyden–Spedicato (ABS) algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as Gram–Schmidt, conjugate direction and Lanczos methods. We present a rank reduction formula for transforming the rows and columns of A, extending the Wedderburn rank reduction formula and the ABS approach. By repeatedly applying the formula to reduce the rank, an extended rank reducing process is derived. The biconjugation process associated with the Wedderburn rank reduction process and the scaled extended ABS class of algorithms are shown to be in our proposed rank reducing process, while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations. The process provides a general finite iterative approach for constructing factorizations of A   and A^T$A^T$ under a common framework of a general decomposition V^TAP=Ω$V^{T}AP=\Omega$. We also show that the biconjugation process associated with the Wedderburn rank reduction process can be derived from the scaled ABS class of algorithms applied to A   or A^T$A^T$. Finally, we provide a list of some well-known reduction procedures as special cases of our extended rank reducing process. The approach is general enough to produce various structured decompositions as well.     

Canonical forms for normal matrices that commute with their complex conjugate

We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate.     

Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.     

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k?2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let G_m(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from G_m(V) onto itself such that for any A,B in G_m(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.     

Normal forms in matrix spaces

For the groupGL(m, C)xGL(n, C) acting on the space ofmxn matrices over C, we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the wild problem of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smith's, Jordan's), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.Partially supported by Israel Science Foundation     

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for

(i)

bilinear forms over an algebraically closed or real closed field;

(ii)

sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and

(iii)

sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F; the canonical matrices are based on any given set of canonical matrices for similarity over F.

A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481-501].     

The Jordan canonical form for a class of zero-one matrices

Let f:N→N be a function. Let A_n=(a_ij) be the n×n matrix defined by a_ij=1 if i=f(j) for some i and j and a_ij=0 otherwise. We describe the Jordan canonical form of the matrix A_n in terms of the directed graph for which A_n is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

On bilinear maps determined by rank one matrices with some applications

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map ? from M_n×M_n to M_n satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,y∈M_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.     

Characterizations of EP matrices and weighted-EP matrices

A square complex matrix A is said to be EP if A and its conjugate transpose A^∗ have the same range. In this paper, we first collect a group of known characterizations of EP matrix, and give some new characterizations of EP matrices. Then, we define weighted-EP matrix, and present a wealth of characterizations for weighted-EP matrix through various rank formulas for matrices and their generalized inverses.