Vandermonde matrices with Chebyshev nodes

For an N×N Vandermonde matrix with translated Chebyshev zero nodes, it is discovered that 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 (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.Similar results hold for VN with translated Chebyshev extreme nodes, too, owing to that admits an explicit QR-like decomposition.Close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method, the minimal residual method, and the generalized minimal residual method on certain linear systems Ax=b with A having eigenvalues the same as the nodes mentioned above. As a by-product, they yield positive definite linear systems for which the residuals by the conjugate gradient method are always comparable to the existing error bounds for all iteration steps.