Interpreting a singular value in a specific problem

Question

In a similar spirit to this post, I pose the following:

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 liter 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 singular value decomposition of the corresponding matrix

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

is

$$A=U\Sigma V^T$$

with

\begin{equation} U=\left[ \begin{array}{cc} -0.9711 & -0.2388\\ -0.2388 & 0.9711 \end{array} \right] \end{equation}

\begin{equation} \Sigma=\left[ \begin{array}{cc} 1.4554 & 0\\ 0 & 0.2199 \end{array} \right] \end{equation}

\begin{equation} V=\left[ \begin{array}{cc} -0.6804 &-0.7329\\ -0.7329 & 0.6804 \end{array} \right] \end{equation}

How do I interpret their physical meaning of the singular values and the columns of the two unitary matrices in the context of this particular problem? That is, what insight do these quantities give me about the nature of the problem itself or perturbations thereof?

Answer 1

I think it is easiest to interpret this when we don't think of the vector spaces in question as $\mathbb{R}^2$, but rather, think of $A$ as a linear map between the vector space $V = (\text{beer}, \text{vodka})$ to the vector space $W = (\text{volume}, \text{alcohol content})$. This map takes the drinks you have, and spits out their corresponding total liquid volume and alcohol content when mixed together.

Now, consider the unit circle in $V$; this is the set of all alcohol drink pairs whose "radius" (which is different from the total liquid volume) is equal to $1$. Now consider the set of all points in $W$ corresponding to the total liquid volumes, alcohol content pairs:

(Ignore the labels above and the angle of the ellipse; it is only a visualization aid)

Then the first left singular vector $u_1$ corresponds to the direction in the $(\text{volume}, \text{alcohol content})$ space that is most sensitive to some change in the $(\text{beer}, \text{vodka})$ space; the amount that it changes by (as a ratio of magnitudes) is given by the singular value $\sigma_1$, and the direction in the $(\text{beer}, \text{vodka})$ space corresponding to this change is the right singular vector $v_1$.

So if you want to use as little beer and vodka as possible (measured in the Euclidean norm rather than the "volume" norm) to affect the greatest change in the corresponding volume and alcohol content of the mixed result, you should add/remove an amount of beer+vodka from your mixture in relative proportions corresponding to the right singular vector $v_1$; in our case, this means adding/removing beer/vodka in a ratio of $0.6804:0.7329$ (the signs don't matter since you can always flip them).

To interpret the second left singular vector $u_2$, we can refer to the diagram again - it corresponds to the next largest change possible in the $(\text{volume}, \text{alcohol content})$ space that is orthogonal to $u_1$. This change has sensitivity given by $\sigma_2$ and is caused by adding/removing beer/vodka in relative proportions given by the second right singular vector $v_2$.