Linear maps that preserve commuting pairs of matrices

Let L be a linear map on the space of n by n matrices with entries in an algebraically closed field of characteristic 0. In this article we characterize all non-singular L with the property that AB = BA implies L(A)L(B) = L(B)L(A).     

Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

Taylor spectral invariance for crisscross commuting pairs on Banach spaces

A theorem on the commuting property of Taylor's spectrum for crisscross commuting pairs is proved in this paper.

     

Linear transformations on symmetric matrices

In this paper we study the problem of characterizing those linear transformations on the vector space of symmetric matrices which preserve a fixed rank or the signature.     

The variety of pairs of commuting nilpotent matrices is irreducible

In this paper we prove the dimension and the irreduciblity of the variety parametrizing all pairs of commuting nilpotent matrices. Our proof uses the connection between this variety and the punctual Hilbert scheme of a smooth algebraic surface.     

Linear transformations on symmetric matrices II

Let Tbe a linear mapping on the space of n× nsymmetric matrices over a field Fof characteristic not equal to two. We obtain the structure of Tfor the following cases:(i) Tpreserves matrices of rank less than three; (ii) Tpreserves nonzero matrices of rank less than K + 1 where Kis a fixed positive integer less than nand Fis algebraically closed; (iii) Tpreserves rank Kmatrices where Kis a fixed odd integer and Fis algebraically closed.     

Linear maps on matrices: the invariance of symmetric polynomials

Let L be a linear map on the space M_n of all n by n complex matrices. Let h(x₁,…,x_n) be a symmetric polynomial. If X is a matrix in M_n with eigenvalues λ₁,…,λ_n, denote h(λ₁,…,λ_n) by h(X). For a large class of polynomials h, we determine the structure of the linear maps L for which h(X)=h(L(X)), for all X in M_n.     

On the problem of describing pairs of commuting Hankel complex matrices

Solutions of two partial problems arising in the course of classifying pairs of commuting Hankel complex matrices are presented.     

Linear operators on matrices: the invariance of the decomposable numerical range

The purpose of this note is to study the structure of all linear operators on matrices which preserve the generalized numerical range. The result generalizes V. J. Pellegrini's determination of all linear operators preserving the classical numerical range in matrix version.     

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ?3, then T(A) = λQAQ^t + Q(A)I_n for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.     

Linear transformations on matrices: rank 1 preservers and determinant preservers

A theorem of Marcus and Moyls on linear transformations on matrices preserving rank 1 and a classical result of Frobenius on determinant preservers are re-proved by elementary matrix methods.     

Linear transformations on symmetric matrices that preserve the permanent

It is shown that if a linear transformation T on the space of n-square symmetric matrices over any subfield of the real field preserves the permanent, where n ? 3, then T(A)= ± PAP^t for all symmetric matrices A and a fixed generalized permutation matrix P with per P= ± 1.     

Solving two subproblems that complete the classification of pairs of commuting Hankel matrices

A new solution of two subproblems completing the classification of pairs of commuting Hankel matrices is proposed, and the minimal orders of matrices solving these subproblems are indicated.