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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.

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    +1 for "getting lost in the forest of symmetric polynomials on $k$ letters" :-)2011-10-17

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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$\dots$

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    Oh, I have time, but do I have the space needed? I'm looking at the $L^2$ scalar products induced by different Kahler metrics on a compact Kahler manifold. On $(1,1)$-forms, one basically gets the Frobenius norm, and on $(p,p)$-forms one gets the Frobenius norm on $p$-th exterior power. So, knowing that you can bound the norm of a $(1,1)$-form lets you bound the norms of its exterior powers, which can for example be helpful in estimating the size of Ricci-tensors of metrics.2011-10-19