How to denote $L_p$ norm of a matrix $X$ in either row or column direction?

Question

How to denote $L_p$ norm of a matrix $X$ in either row or column direction? The result of such an operation will be a column or row vector. Representing $\| X \|_p$ is ambiguous, isn't it? Do you have any suggestion for such a notation?

Answer 1

For a matrix $A$ write the elements as $a_i^j$ for $i$ and $j$ the row and column index respectively.

Write $\|A\|^p$ for the vector of all row $L_p$ norms. Then $\|A\|^p$ is a column vector. It does not have a column index and you can write things like $\|A\|^p_i$ for the $L_p$ norm of row $i$ of the matrix.

Likewise Write $\|A\|_p$ for the vector of all column $L_p$ norms and write things like $\|A\|^j_p$ for the $L_p$ norm of column $j$ of the matrix.

Usually this would create ambiguity since we might take the $p$-th power of the $p$-norm of a vector. Meaning $\|x\|_p^p = \sum|x_i|^p$. However in your case $\|A\|_p$ is a vector and we will never take the $p$-th power of a vector.

The only ambiguity I see is if you also write $\|A\|$ to mean something. For example the operator norm. But I don'r see why you would take the $p$-th power of that.

You could also put the indices on the left and write $_p\|A\|$ and $^p\|A\|$ if you really want to avoid ambiguity.