Given N numbers α1, α2,..., αN called nodes, the associated Vandermonde Matrix is d...

Submitted by V. Mehrmann For an N × N Vandermonde matrix VN = (αi−1j)1ijN with translated Chebyshev zero nodes, it is discovered thatV T N admits an explicit QR decomposition with the R-factor consisting of the coefficients of the translated Chebyshev polynomials. This decomposition then leads to an exact expression for the Frobenius condition number of its submatrix Vk,N = (αi−1j)1ik,1jN (so-called a rectangular Vandermonde ma-trix), bounds on individual singular value, and more. It is explained how these results can be used to establish asymptotically optimal lower bounds on condition numbers of real rectangular Vandermonde matrices and nearly optimally conditioned real rectangular Vandermonde matrices on a given interval. Extensions are also made for VN with nodes being zeros of any translated orthogonal polynomials other than Chebyshev ones. Similar results hold for VN with translated Chebyshev extreme nodes, too, owing to that V TN admits an explicit QR-like decomposition. Close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method, th