  We consider symmetric positive definite preconditioners for multiple
saddle-point systems of block tridiagonal form, which can be applied within the
MINRES algorithm. We describe such a preconditioner for which the
preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the
preconditioner is applied exactly. We discuss the relative merits of such an
approach compared to a more widely studied block diagonal preconditioner,
specify the computational work associated with applying the new preconditioner
inexactly, and survey a number of theoretical results for the block diagonal
case. Numerical results validate our theoretical findings.
