Maximum "stretch factor" of linear map

In general, the maximum stretch factor is the largest singular value $\sigma_1$ of $A$, i.e. the square root of the largest eigenvalue $\lambda_1$ of $A^TA$. Note that if you assume $A$ is symmetric, then $\sigma_1$ is exactly $\sqrt{\lambda^2} = |\lambda|$ where $\lambda$ is the largest eigenvalue of $A$.

This comes from our knowledge of quadratic forms. Since maximizing $\|Ax\|$ is equivalently maximizing the square root of $\|Ax\|^2 = (Ax)\cdot(Ax) = x^TA^TAx$, we’re really maximizing the square root of the quadratic form given by $A^TA$. The restriction $\|x\|=1$ tells us that this is exactly the largest singular value of $A$, $\sigma_1$.

Note that if $x$ is the eigenvector for $A^TA$ corresponding to $\lambda_1$, then

$$\sqrt{x^TA^TAx} = \sqrt{x^T(\lambda_1x)} = \sqrt{\lambda_1\|x\|^2} = \sigma_1\|x\|.$$

Thus, if you want to maximize the length of vectors on a sphere of any fixed radius, i.e. constraining yourself to $\|x\| = r$ for a fixed $r$, you just scale by $r$.