Canonical variances

The $v \times v$ symmetric, nonnegative definite matrix $C_d$ is never of full rank; its maximal rank is $v-1$, which is achieved exactly when the block design $d$ is connected. Denote the $v-1$ ordered, largest eigenvalues of $C_d$ by

$$x_{d1} \leq x_{d2} \leq \cdots \leq x_{d,v-1}$$

Design $d$ is connected if and only if $x_{d1} >0$. The corresponding nonzero eigenvalues of $C^{+}_d$ are the inverses of the nonzero $x_{di}$'s ; for a connected design these are

$$z_{d1} \geq z_{d2} \geq \cdots \geq z_{d,v-1}$$

The $z_{di}$ 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 $C_d$ orthogonal to the all-ones vector. We define a full set of $v-1$ canonical variances even for disconnected designs, in which case some of the $z_{di}$ 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 $x_{di}$'s.