Relating the singular values of a scaled matrix to its determinant

Question

Let $A$ be a real, $n\times n$,full rank matrix with singular values: $\sigma_1\ge\dots \ge \sigma_n$. Assuming the rows of $A$, $a_1,\dots,a_n$ are scaled so that $\|a_i\|_2 = 1$ for $i=1,\dots,n$, what bounds can be made, if any, on the singular values of $A$ related to $\det(A)$? Specifically can we get a lower bound on $\sigma_1$ and $\sigma_n$ using $\det(A)$? The intuition (perhaps wrong) being that the determinant can be related to the volume of the norm $1$ ball under the transformation defined by $A$ and $\sigma_1 ,\dots, \sigma_n$ can be used to relate the norm $2$ ball before and after applying the transformation $A$.

Answer 1

Sure; from taking the determinant of both sides of the SVD we have that $|\det A| = \Pi_i \sigma_i.$ So $$\sigma_1 \geq |\det A|^{1/n}.$$

This bound is tight since equality holds in the case that $A$ is a multiple of the identity matrix.