Explaining the physical meaning of an eigenvalue in a real world problem

Question

Contextual Problem

A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the next day. Since all committee members will be driving home immediately after his defense, he wants to make sure that they all drive home safely. To do so, he must ensure that his mixture doesn't contain too much alcohol in it!

Therefore, his goal is to make a 10 gallon mixture of vodka and beer such that the total alcohol content of the mixture is only $12$ percent. Suppose that beer has $8\%$ alcohol while vodka has $40\%$. If $x$ is the volume of beer and $y$ is the volume of vodka needed, then clearly the system of equations is

\begin{equation} x+y=10 \\ 0.08 x +0.4 y = 0.12\times 10 \end{equation}

My Question

The eigenvalues and eigenvectors of the corresponding matrix

\begin{equation} \left[ \begin{array}{cc} 1 & 1\\ 0.08 & 0.4 \end{array} \right] \end{equation}

are

\begin{align} \lambda_1\approx 1.1123 \\ \lambda_2\approx 0.2877 \\ v_1\approx\left[\begin{array}{c} 0.9938 \\ 0.1116 \end{array} \right] \\ v_2\approx\left[\begin{array}{c} -0.8145 \\ 0.5802 \end{array} \right] \end{align}

How do I interpret their physical meaning in the context of this particular problem?

Answer 1

An interpretation of eigenvalues and eigenvectors of this matrix makes little sense because it is not in a natural fashion an endomorphism of a vector space: On the "input" side you have (liters of vodka, liters of beer) and on the putput (liters of liquid, liters of alcohol). For example, nothing speaks against switching the order of beer and vodka (or of liquid and alcohol), which would result in totally different eigenvalues.

Answer 2

First we need to interpret the transformation "physically". What this does is gets the amount of each type of alcohol as input, and spits out the total volume and alcohol volume as output.

I'd be surprised if this has a serious "physical interpretation", because the input and output are of different types. I suppose I'd say that the eigenvectors are the alcohol mixes where the ratio of beer to vodka is the same as the percentage of alcohol in the final mixture. Technically interprable, but probably no more than a curiosity.

Answer 3

While the eigenvectors and eigenvalues don't play their usual role in this problem (as argued in the other answers) the eigensystem still has a physical interpretation.

The eigenvector $v_2$ is unphysical since it corresponds to a negative volume.

Let $M$ represent the matrix above, $$M v_1 = \lambda_1 v_1.$$ Since $\lambda_1 y/(\lambda_1 x) = y/x$, the alcohol content of the final mixture is equal to the ratio of vodka to beer. In addition $$x+y = \lambda_1 x,$$ so the eigenvalue is the alcohol content of the final mixture plus one.

This mixture could be used since the alcohol content is about 11 percent. To get a 10 liter mixture scale the eigenvector, $v_1 \rightarrow 10v_1/(x+y)$. Perhaps the committee would find it a more mathematically interesting mixture.

Answer 4

You don't really need eigenvectors to solve this problem. Just treat it as a linear programming problem and let the Simplex Algorithm (or your algorithm of choice) come up with a feasible solution. Since you need an objective function, you can choose one arbitrarily; for example, you might choose to minimize the total cost. If you don't know the costs of beer and vodka, just make some up.

Answer 5

Problem is very simple and interesting λ1 ≈ 1.1123 is content of pure beer alcohol volume vise and λ2≈0.2877 is content of pure vodka alcohol volume wise ($8%$ beer in one liter = $80 \mathrm{mL}$ pure beer alcohol).

In Volume direction has no significance so here eigenvector are pure mathematical term.