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

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.


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.


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.

Tags:

Eigenvalues Eigenvectors

Related

Source to learn Galois Theory Is B(H) a Hilbert space? An integral of a modified Bessel function: $\int_{0}^{\infty} K_{0} \left(\sqrt{a(k^{2}+b)} \right) dk$ Can $\sum_{x \in [0,1]} e^x$ be represented as an integral? If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$ For any set $A\subseteq\mathbb{R}^n$, we have $ \overline{A^{\circ}} = \overline{\overline{A^{\circ}}^{\,\circ}}$ Logical systems that are complete but not sound What are the differences between a claim, a statement and a proposition? Why does one counterexample disprove a conjecture? Is it possible to divide a circle into $7$ equal "pizza slices" (using geometrical methods)? Geometric understanding of differential forms.

Recent Posts