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.