Let $A$ be an $n \times n$ matrix over the complex numbers. The Frobenius norm of $A$ is defined by
$ \| A \| = Tr(A \cdot A^*) $
where $A^*$ is the conjugate transpose of $A$.
Now let $\wedge^k A$ be the matrix of $k \times k$ minors of $A$. Then we have a Frobenius norm of $\wedge^k A$, given by
$ \| \wedge^k A \| = Tr(\wedge^k A \cdot \wedge^k A^*). $
I would like to majorize the norm of $\wedge^k A$ by the one of $A$, and I'm fine with doing this very brutally. So, for example, is there a constant $C$ such that $\| \wedge^k A \| \leq C \| A \|^k$? It seems like that inequality should hold with $C = 1$, but I keep getting lost in the forest of symmetric polynomials on $k$ letters when I try to prove it.