[en] We prove two theorems on the simultaneous diagonalizability of a set of complex square matrices by a biunitary transformation. (Author)

[en] There are many theorems concerning the eigenvalues of matrices with dominant main diagonal, but very few when this condition is not satisfied. The theorem here proposed belongs to this second category

[en] Matrix orderings on rings are investigated. It is shown that in the commutative case they are essentially positive cones. This is proved by reducing it to the field case; similarly one can show that on a skew field, matrix positive cones can be reduced to positive cones by using the Dieudonne determinant. Our main result shows that there is a natural bijection between the matrix positive cones on a ring R and the ordered epic R-fields. (author). 7 refs

[en] Let matrices A,C ∈ M_n have eigenvalues α₁, . . ., α_n and γ₁, . . . , γ_n, respectively. The set of complex numbers D_C(A) = {det(A−UCU*) : U ∈ M_n, U*U = I_n} is called the C-determinantal range of A. The paper studies various conditions under which the relation D_C(R S) = D_C(S R) holds for some matrices R and S.

[en] We study the commutativity of certain class of rings, namely rings with unity 1 and right s-unital rings under each of the following properties [yx^m - xⁿ f(y)x^p,x]=0, [yx^m+xⁿf(y)x^p,x]=0, where f(t) is a polynomial in t²Z[t] varying with pair of ring elements x,y and m,n,p are fixed non-negative integers. Moreover, the results have been extended to the case when m and n depend on the choice of x and y and the ring satisfies the Chacron's Theorem. (author). 14 refs

[en] Decomposition of unipotents gives short polynomial expressions of the conjugates of elementary generators as products of elementaries. It turns out that with some minor twist the decomposition of unipotents can be read backwards to give very short polynomial expressions of the elementary generators themselves in terms of elementary conjugates of an arbitrary matrix and its inverse. For absolute elementary subgroups of classical groups this was recently observed by Raimund Preusser. I discuss various generalizations of these results for exceptional groups, specifically those of types E₆ and E₇, and also mention further possible generalizations and applications.

[en] Positive-definite Hankel matrices have an important property: the ratio of the largest and the smallest eigenvalues (the spectral condition number) has as a lower bound an increasing exponential of the order of the matrix that is independent of the particular matrix entries. The proof of this fact is related to the so-called Vandermonde factorizations of positive-definite Hankel matrices. In this paper the structure of these factorizations is studied for real sign-indefinite strongly regular Hankel matrices. Some generalizations of the estimates of the spectral condition number are suggested

[en] The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries ${\mathcal{A}}_{i,j}$ and ${\mathcal{A}}_{j,i}$ are real and have opposite signs, or are both zero, and the diagonal is zero. This generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology. (paper)

[en] Let A be an n by n matrix which may be singular with a one-dimensional null space, and consider the LU-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero nth pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is O(sigma/sub n/) or O(kappa^-1(A)), where sigma/sub n/ is the smallest singular value of A and kappa(A) is the condition number of A. These conditions are expressed in terms of the elements of A^-1 in general but reduce to conditions on the elements of the singular vectors corresponding to sigma/sub n/ when A is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is guaranteed to produce a small nth pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix

[en] This paper was motivated by the question: Does, like the ring case, the process of taking Jacobson Radicals and constructing matrix near rings coincide? We present here, for the class of weakly distributive d.g. near rings, a conditionally affirmative answer to this question. Our main result is as follows: Let R be a weakly distributive d.g. near ring with identity. If J(R) is contained δ₁M_n(R), the derived group of M_n(R), then J(R)=J(M_n(R)) where I-bar is defined as a subnear ring of M_n(R) generated by {f^a_ij; a is an element of I, 1 ≤ i, j ≤ n}. (author). 6 refs