4
$\begingroup$

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:

$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\\infty ~~~\text{otherwise}\end{cases}$$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.

My question is:

Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?

$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.

$\endgroup$
0

1 Answer 1

7
$\begingroup$

Let $p^*$ and $q^*$ be the conjugate exponents. Some (slightly laborious) algebra shows that the dual-norm is $\|A\|_{p^*,q^*}$. The conjugate function is the indicator function for the (unit) dual-norm ball.

$\endgroup$
1
  • $\begingroup$ This has also been shown in much more generality in the context of (weighted) Besov (sequence) spaces (characterised by wavelets) - which are a vast generalisation matrix spaces endowed with $\| \cdot \|_{p, q}$ - in the paper On the smoothness and convexity of Besov spaces by Kazimierski. $\endgroup$
    – Ramanujan
    Mar 8 at 21:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.