The symmetric, nonnegative definite matrix is never of full rank; its maximal rank is , which is achieved exactly when the block design is connected. Denote the ordered, largest eigenvalues of by
Design is connected if and only if . The corresponding nonzero eigenvalues of are the inverses of the nonzero 's ; for a connected design these are
The are called the canonical variances. They are the variances of a set of contrasts whose vectors of coefficients are any orthonormal set of eigenvectors of orthogonal to the all-ones vector. We define a full set of canonical variances even for disconnected designs, in which case some of the are taken as infinity. An infinite canonical variance corresponds to a contrast which is not estimable.
Many of the commonly used design optimality criteria are based on the canonical variances. Because of their importance they have merited an element, canonical_variances, in the external representation. Infinite values are recorded there as ``not_applicable'' and, as already explained, correspond to zero values of 's.