A condition for the convexity of the norm-numerical range of a matrix

AbstractLet v be a norm on Cn, and let a be a matrix which has a v-Hermitian decomposition. (1) The v-numerical range of a is convex. (This generalizes the Hausdorff-Toeplitz theorem.) In fact, the v-numerical range is equal to the field of values of a matrix similar to a. (2) If the Hermitian and v-Hermitian decompositions of a coincide, then the v-numerical range of a and the field of values of a are the same. This follows from detailed information about the boundary of the range

Transitive closure and related semiring properties via eliminants

AbstractClosed semirings are algebraic structures that provide a unified approach to a number of seemingly unrelated problems of computer science and operations research. For example, semirings can be used to describe the algebra related to regular expressions, graph-theoretical path problems, and linear equations. We present a new axiomatic formulation of closed semirings. We introduce the concept of eliminant, which simplifies the treatment of closed semirings considerably and yields very simple proofs of otherwise difficult theorems. We use eliminants to define matrix closure, formulate closure algorithms, and prove their correctness

Fraction Free Gaussian Elimination for Sparse Matrices

AbstractA variant of the fraction free form of Gaussian elimination is presented. This algorithm reduces the amount of arithmetic involved when the matrix has many zero entries. The advantage can be great for matrices with symbolic entries (integers, polynomials, expressions in trigonometric functions, etc.). These claims are supported with some analysis and experimental data