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I have a problem below:

Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$.

Now suppose that $H: \mathbb{R}^k \rightarrow \mathbb{R}^k \times \mathbb{R}^d$ is a matrix-valued function such that $\|H\|=O(\|x\|).$ What can we say about $\|H^T\|$? Thanks for any help.

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The norm of every matrix is bounded. – copper.hat Jan 6 '14 at 7:45
Thanks @copper.hat, I think I misunderstood. But, I've corrected now. – Jlamprong Jan 6 '14 at 7:54
up vote 2 down vote accepted

All norms on $\mathbb{R}^n$ are equivalent (and on $\mathbb{R}^m \times \mathbb{R}^n$ as well, of course).

So you can use the Frobenius norm, and since $\|H\|_F = \|H^T \|_F$, whatever you can say about $H$, you can say about $H^T$ (with appropriate constant factors, of course).

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