How can I find the maximum values of a transformation of the unit sphere?

Question

I'm interested in the maximum values possible in each dimension when multiplying the unit circle/sphere by a rectangular or square transformation matrix. For a small matrix I can solve the values by hand with calculus. Is there a solution for a generic matrix? E.g. find the maximum values $y_1$ and $y_2$ where $\sqrt{x_1^2 + x_2^2} = 1$ $$\begin{bmatrix}y_1\\y_2\end{bmatrix} = \mathbf{A} \begin{bmatrix}x_1\\x_2\end{bmatrix}$$ example points for $A = \begin{bmatrix}1&3\\4&2\end{bmatrix}/4$

Answer 1

Yes, this is simple to compute. Notice that

$$y_1 = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \end{bmatrix} \mathbf{A} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.$$

Denoting the projection $P_1 := \begin{bmatrix} 1 & 0 \end{bmatrix}$, then, you're looking for the maximum output of $P_1 \mathbf{A} \mathbf{x}$ for $\|\mathbf{x}\| =1$. Notice $P_1 \mathbf{A}$ is simply the row vector that is the first row of $\mathbf{A}$, so the Cauchy-Schwartz inequality with respect to the standard inner product on $\mathbb{R}^2$ ensures the quantity you're looking for is $\|P_1 \mathbf{A}\|$, computed as a vector again according to the standard inner product. So the quantities you're looking for are simply the magnitudes of the rows of $\mathbf{A}$. This yields $\frac{\sqrt{10}}{4}$ and $\frac{\sqrt{5}}{2}$ for your example matrix.