We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. W...

The capacity of symmetric, neighboring and consecutive side-information single unicast index coding problems (SNC-SUICP) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar. For these index coding problems, an optimal index code construction by using Vandermonde matrices was proposed. This construction requires all the side-information at the receiver...

In this paper, we give a matrix decomposition method used to calculate unified generalized Stirling numbers in an explicit, non-recursive mode, and some of its applications. Then, we define generalized factorial matrices which may be regarded as a generalization in the form of the Vandermonde matrices, and presents some of their properties — in particular, triangular matrix factors of the inver...

We show that certain matrices built from Vandermonde matrices are of full rank. This result plays a key role in the construction of the “limit theory of generic polynomials”.

Multipoint polynomial evaluation and interpolation are fundamental for modern algebraic and numerical computing. The known algorithms solve both problems over any field by using O(N log N) arithmetic operations for the input of size N , but the cost grows to quadratic for numerical solution. Our study results in numerically stable algorithms that use O(uN log N) arithmetic time for approximate ...

We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize existing nonsingular arbitrary rank. For totally nonnegative above classes, can be computed efficiently high relative accuracy componentwise in floating point arithmetic. In turn, matrix c...

This paper deals with Vandermonde matrices V whose nodes are the equidistant points in [0,1]. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both V and its inverse and show that a new representation of the floating point number system allows one to build an ...

This work examines various statistical distributions in connection with random Vandermonde matrices and their extension to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be O(logN) and O(logN/ log logN) respectively where N is the dimension of the matrix, generalizing the results in [21]. We further study the behavior of the mini...

We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible.