A circuit is a closed loop, i0,...,in−1,in = i0 of length n ≥0. Conservation
of energy is a condition on the k ×k matrix V to the effect that, on each circuit the sum is zero
V (i0,i1) + V (i1,i2) + ···+ V (in−1,i0) = 0.
A matrix satisfying this condition is called conservative. Show that each conservative matrix is skew-
symmetric. Deduce that the set of conservative matrices is a vector space, closed under vector-space op-
erations. Exhibit a 3 ×3 skew-symmetric matrix that is not conservative. A skew-symmetric matrix of the
form V (i,j) = αi −αj is called additive. Prove that every conservative matrix is additive. What is the
dimension of the vector space of conservative 6 ×6 matrices?
The following exercises refer to the linear model for the voltages in which E(Yij ) = αi −αj is conservative.
The data is as follows:
(a) For a single k×k table, obtain an expression for the least-squares estimate of α= (α1,...,α5). Use this
formula to compute ˆαfor each of the three electrolytes. Explain why (α1,...,α5) and (α1,...,α5) +
(c,c,c,c,c) are equivalent as parameter points in the model.
(b) Assess the evidence for and against the hypothesis that the vector of potentials is constant across elec-
trolytes. That is to say, fit the linear model in which the potentials are constant across electrolytes, and
compare the fit with the model in which α varies from one electrolyte to another. Obtain the relevant
sums of squares, their degrees of freedom, and compute the appropriate F-statistic.
(c) Discuss briefly the arguments for and against analysis of these data by linear models after transformation.