TITLE: Is the Frobenius norm on exterior powers majorized by a power of the norm on matrices?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: I answer your first question.
The map $(A_1,\dots A_k)\in (End(\mathbb{C}^n))^{\times k}\mapsto A_1\wedge\dots\wedge A_k\in End((\mathbb{C}^n)^{\wedge k})$  is multilinear between finite-dimensional Banach spaces so it is automatically continuous.
The continuity of a multilinear map is the same as the boundedness, i.e. there exists a real number $C_k$ such that $\|A_1\wedge\dots\wedge A_k\|\leq C_k\|A_1\|\dots\|A_k\|$.  
How to estimate such bounds $C_k$ is another question$\dots$