Some notes on de Oliveira's determinantal conjecture (original) (raw)

On determinant preserver problems

Linear Algebra and its Applications, 2003

Let M n and T n be the vector spaces of n × n matrices and upper triangular matrices over a field F (with some cardinality and characteristic restrictions) respectively. We characterise transformations φ on these two spaces separately which satisfy one of the following conditions: 1. det(A + λB) = det(φ(A) + λφ(B)) for all A, B and λ. 2. φ is surjective and det(A + λB) = det(φ(A) + λφ(B)) for all A, B and two specific λ. 3. φ is additive and preserves determinant.

On real determinantal quartics

2010

We describe all possible arrangements of the ten nodes of a generic real determinantal quartic surface in P 3 with nonempty spectrahedral region.

Towards the Casas- Alvero conjecture

2015

We investigate necessary and sufficient conditions for an arbitrary polynomial of degree nnn to be trivial, i.e. to have the form a(z−b)na(z-b)^na(z−b)n. These results are related to an open problem, conjectured in 2001 by E. Casas- Alvero. It says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. In particular, we establish determinantal representation of the Abel-Goncharov interpolation polynomials, related to the problem and having its own interest. Among other results are new Sz.-Nagy type identities for complex roots and a generalization of the Schoenberg conjectured analog of Rolle's theorem for polynomials with real and complex coefficients.

On the C-determinantal range for special classes of matrices

Applied Mathematics and Computation, 2016

Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det (A − U CU *) : U U * = I n }. If A, C are both Hermitian matrices, then by a result of M. Fiedler [11] this set is a real line segment. In this paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to C.-K. Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, see [21, Theorem 3.3]. C (A) = V CV * (U AU *) for any U, V ∈ U n. Definition 1.2. The σ-points of C (A) are defined by z σ = n i=1 (α i − γ σ(i)), σ ∈ S n , where α 1 ,. .. , α n and γ 1 ,. .. , γ n are the eigenvalues of A and C, respectively. It is easy to see that all the (not necessarily distinct) n! σ-points belong to C (A). The characterization of the C-determinantal range of A for Hermitian matrices A and C was obtained by M. Fiedler [11], who proved that C (A) is a real line segment, whose endpoints are the minimal and maximal σ-points of C (A). The C-determinantal range of A is intimately connected with a famous conjecture of M. Marcus [22] and G. N. de Oliveira [24], which can be reformulated as follows: for normal matrices A, C ∈ M n it holds that C (A) is a subset of the convex hull of the σ-points z σ , σ ∈ S n. This

A note on finite determinacy of matrices

arXiv: Algebraic Geometry, 2020

In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by change of coordinates together with multiplication by invertible matrices from both sides.

Three observations on the determinantal range

Linear Algebra and its Applications, 2005

Let A, C ∈ M n , the algebra of n × n complex matrices. The set of complex numbers C (A) = {det (A − U CU *) : U * U = I n } is the C-determinantal range of A. In this note, it is proved that C (A) is an elliptical disc for A, C ∈ M 2. A necessary and sufficient condition for C (A) to be a line segment is given when A and C are normal matrices with pairwise distinct eigenvalues. The linear operators L that satisfy the linear preserver property C (A) = C (L(A)), for all A, C ∈ M n , are characterized.

MORE CALCULATIONS ON DETERMINANT EVALUATIONS

The purpose of this article is to prove several evaluations of determinants of matrices, the entries of which are given by the recurrence a i,j = a i−1,j−1 +a i−1,j , i, j ≥ 2, with various choices for the first row a 1,j and first column a i,1 .