J.A. Dias da Silva

On the eigenvalues of the matrix

Let A A be n×n matrices. We study the eigenvalues of when X runs over the set of n×n nonsingular matrices.

On the μ-colorings of a matroid

Let Mbe a matroid on S. Let μ = (μ1,⋯, μK) be a partition of ∣S∣ (:∣ ∣ denotes cardinality). We state a necessary and sufficient condition for the existence of pairwise disjoint independent sets of M I 1, [cddot]I 1, satisfying the following conditions. We use this result to restate a theorem of CGamas [2], giving necessary and sufficient conditions for nonvanishing of decomposable symmetrized tensors.

The perron root of a weighted geometric mean of nonneagative matrices

For k nonnegative n-by-nmatrices A 1…A k we consider the matrix where is the (entry-wise) Hadamard product and is the component-wise weighted geometric mean of It is shown that for the inequality holds. Here p denotes the spectral radius. The case of equality is characterized and it is shown that p(C), considered as a function of α = (α1,⋯,αk), is convex. This generalizes recent results of Schwenk, and of Karlin-Ost. Similarly, we consider for A ⩾ 0 the comparison matrix M(A). where M(A) ij = aij for i = j, and = -aij for i ≠ j. If ω(A) denotes the minimal real eigenvalue of M(A) then it is shown that if ω(Ai ) > 0i = 1, …, K and the dual inequality, holds. Certain other inequalities, some already known, are related to these, and several characterizations are given for another quantity associated with a nonnegative matrix.

Equality of decomposable symmetrized tensors and ∗-matrix groups

Let G be a subgroup of the full symmetric group Sn, and χ a character of G. A ∗-matrix can be defined as an n × n matrix B which satisfies dGx(BX) = dGx(X) for every n × n matrix X. They form a multiplicative group, denoted S(G,X), which plays a fundamental role in the study of equality of two decomposable symmetrized tensors. The main result of this paper (Theorems 2.2, 2.3, and 2.4) is a complete description of the matrices in S(G,X). This description has many consequences that we present. There are also results on related questions.

Nonzero star products

Many authors have considered the problem of determining necessary and sufficient conditions for the star product of mvectors to be zero. Probably the most remarkable results are those of Merris [9] and Gamas [4]. Our purpose is to generalize these two results to results arbitray symmetry classes of tensors.

On the multilinearity partition of an irreducible character

The concept of k-dimension of a family of vectors is introduced. Using this concept we are able to define the k-index of a symmetry class of tensors, generalizing the Marcus—Chollet index. Both notions enable us to associate to each irreducible character of a subgroup G of Sm a partition of m. It is also proved that if G=Sm this partition coincides with the partition usually associated with the irreducible characters of Sm .

Conditions for equality of decomposable symmetric tensors

Abstract Let x1,…,xm be linearly independent vectors. We give a necessary and sufficient condition for x 1 ∗…∗x m =y 1 ∗…∗y m to hold. Some consequences of this result are also presented.

On simultaneous similarity of matrices and related questions

Abstract Let F be a field and F[x] the ring of polynomials in an indeterminate x over F . Let M n (F), M n (F[x]) denote the algebras of n × n matrices over F, F[x] , respectively, and GL(n,F), GL(n,F[x]) their corresponding groups of units. Given A(x), B(x) ∈ M n (F[x]) , we say that A(x), B{x) are PS -equivalent ( = “polynomial-scalar”) if there exist P(x) ∈ GL(n,F[x]), Q ∈ GL(n,F) with B(x) = P(x)A(x)Q . We consider the problem of determining whether A(x) and B(x) are PS -equivalent. In other words we wish to classify the orbits of M n (F[x]) under the action of GL(n,F[x]) × GL(n,F) acting via (T(x),Q)A(x) = T(x) −1 A(x)Q . We observe that the classical problems of determining the simultaneous equivalence of two k -tuples of elements of M n (F) and the simultaneous similarity of two k -tuples of elements of M n (F) are special cases of this problem. We observe that the Smith invariants of A(x) and B(x) (that is, invariants for the action of GL(n,F[x]) × GL(n,F[x]) on M n (F[x]) via ( T(x),S(x))A(x) = T(x) −1 A(x)S(x)) must be equal if A(x), B(x) are PS -equivalent. Based on this we present a near canonical form for PS -equivalence and an algorithm for determining whether two matrices in near canonical form are PS -equivalent. We examine in detail the “generic case” in which A(x) has a single Smith invariant different from 1 and obtain a further set of invariants in this case, and based on these we present an improved algorithm, to determine PS -equivalence in this situation. While the main emphasis in the paper is on finding a reasonably good algorithm in the generic ease, we also discuss the question of finding a complete set of invariants for PS -equivalence, especially in the case n = 2. where connections with linear fractional transformations arise. A much more comprehensive account of the invariants in the simultaneous similarity problem can be found in Friedlands paper (Adv. in Math. 50 (1983) 189–265).

Conditions for equality of decomposable symmetric tensors. II.

Abstract We derive consequences of a condition for the equality of two star products given by the second author. We also study another method for the same problem which consists of comparing the components, in an appropriate basis, of the star products involved.

On the Schur inequality

We generalize to a large class of functions the conditions of equality in the Schur inequality.