Optimal measures for Toeplitz matrices

We extend and refine a result of R. McEachin concerning 2 × 2 Toeplitz matrices. Even in the complex case, the Schur norm of such a matrix is equal to the total variation of an appropriately chosen representing measure. The measure may be supported on two or three points. From this phenomenon we derive formulas for the Schur norms of the matrices. We point out a connection with the extreme points of the unit ball in a certain function space.