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) +...

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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.
Answered 9 days AfterApr 27, 2022

Answer To: A circuit is a closed loop, i0,...,in−1,in = i0 of length n ≥0. Conservation of energy is a...

Robert answered on May 06 2022
77 Votes
Solutions
Ans 1.
A) Each conservative matrix is skew symmetric as explained below:
Any matrix can be
skew symmetric if and only if it is opposite of its transpose and all the main diagonal entries of a skew symmetric matrix are zero.
i.e., AT = -A
or aji = - aij
for example:
Similarly, as given in the question that in the closed loop circuit with i0, i1---------in (where in = i0) the sum total of voltage in each loop is 0 and it is satisfying conservative condition. It can be written as below:

Here, also the sum of main diagonal is zero means it is showing skew symmetry or a skew symmetric matrix. Since all the conservative matrix has closed loop circuit, they also have the sum of their main diagonals equal to zero and this shows that each conservative matrix is skew symmetric.
B) The set of conservative matrices is a vector space, closed under vector space operations as explained below:
Conservative matrix is a skew symmetric matrix and each skew symmetric matrix has two properties –
a) The sum...
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