A matrix satisfying this condition is called conservative.Show that each conservative matrix is skew- symmetric

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

2. A circuit is a closed loop, io,...,in-1, in ip of length n ≥ 0. Conservation

of energy is a condition on the k x k matrix V to the effect that, on each circuit the sum is zero

V(i0, i1) + V(i1, 12) + ··· +V(in-1, 0) = 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 x 3 skew-symmetric matrix that is not conservative. A skew-symmetric matrix of the form V(i,j) = a;a; is called additive. Prove that every conservative matrix is additive. What is the dimension of the vector space of conservative 6 x 6 matrices?

The following exercises refer to the linear model for the voltages in which E(Y1j) = a; -a; is conservative. The data is as follows:

Electrolyte O

Mg

Zn

Fe

Pb

Cu

Mg Zn

0.0

0.414

0.807

0.876

1.291

-0.414

0.0

0.429

0.533

0.886

225

Fe

-0.807

-0.429

0.0

0.043

0.377

Pb

-0.876

-0.533

-0.043

0.0

0.271

Cu

-1.291

-0.886

-0.377

-0.271

0.0

Electrolyte A

Mg

0.0

0.247

0.856

1.051

1.402

Zn

-0.247

0.0

0.434

0.521

0.867

Fe

-0.856

-0.434

0.0

0.058

0.443

Pb

-1.051

-0.521

-0.058

0.0

0.374

Cu

-1.402

-0.867

-0.443

-0.374

0.0

Electrolyte K

Mg

0.0

0.443

0.895

0.973

1.281

3228

Zn

-0.443

0.0

0.477

0.503

0.856

Fe

-0.895

-0.477

0.0

0.107

0.432

Pb

-0.973

-0.503

-0.107

0.0

0.392

Cu

-1.281

-0.856

-0.432

-0.392

0.0

(a) For a single kx k table, obtain an expression for the least-squares estimate of a = (a1,...,05). Use this formula to compute â for each of the three electrolytes. Explain why (a1,..., as) and (a1,...,05) + (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 a 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.

 

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