Vandermonde matrices with Chebyshev nodes Original Research Article

For an N×N Vandermonde matrix image with translated Chebyshev zero nodes, it is discovered that image 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 image (so-called a rectangular Vandermonde matrix), 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.