I was wondering whether there is a way to obtain the determinant of a matrix out of its norm (when the matrix is regular otherwise it is not true). If $A$ is a square matrix of dimension $n\geq 1$, and $\det A\neq 0$, do we have something like?
$$|Ax|_2 \leq \|A\|_{\text{op}} |x|_2 \leq C_n |\det A| \ |x|_2 \leq \cdots$$
or similar? Or maybe a similar estimate for different norms for $A$ and $x$ if needed since many matrix norms are related to eigen- or singular values which are related to the determinant.
Thanks a lot! 🙂
Best Answer
Consider the matrices of the form $\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right)$, with $x\in(0,+\infty)$. All of them have determinant equal to $1$, but their norms are arbitrarily large. Does this answer your question?