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 $\mathbf{A}$.


1 Answer 1


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.

  • $\begingroup$ 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

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