We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-ha...

Lower bounds on the condition number κp(Vc ) of a real confluent Vandermonde matrix Vc are established in terms of the dimension n or n and the largest absolute value among all nodes that define the confluent Vandermonde matrix or n and the interval that contains the nodes. In particular, it is proved that for any modest kmax (the largest number of equal nodes), κp(Vc ) behaves no smaller than ...

We consider a certain decomposition of the matrix algebra Mn(F ), where F is a field. The commutation relations of that decomposition yield an n × n matrix Mn , which determines the multilinear polynomial identities of Mn(F ). Thus if char(F ) = 0, the matrix Mn ) determines the polynomial identities of Mn(F ). We show that M Mn(F ) is very close to the tensor product of two n × n Vandermonde m...

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester’s formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determi...

We derive a closed form for the Jordan decomposition of companion matrices including properties of generalized eigenvectors. As a consequence, we provide a formula for the inverse of confluent Vandermonde matrices and results on sensitivity of multiple roots of polynomials.

Threshold schemes are ideally suited to situations where a group of mutually suspicious individuals with conflicting interests must cooperate. The schemes allow a user to divide portions of a secret among a participants group. Any t or more participants from a participants group of n members can cooperate to regain the original secret while (t-1) or fewer participants cannot reveal anything abo...