Does there exists commonly used ( possible partial) orderings which would rank matrices as a function of their "degree of symmetry"?
I am thinking one could for instance have $\succeq_{SYM}$ defined as :
For any two matrices $A,B$ with the same dimensions, if $|a_{ij} -a_{ji}| \leq |b_{ij} - b_{ji}|$ for every $i,j$, then $A \succeq_{SYM} B$.
Does this partial ordering has a commonly used name? Do you know of others which would be in use?
That ordering sounds very basis-dependent. How about something based on the fact that exactly the symmetric matrices are orthogonally diagonalizable? For example, define the "asymmetry" of $A$ as $$ \operatorname{asym} A = \inf_{P\in\mathrm{SO}(n), \Delta\text{ diagonal}} \|P^{\sf t}AP-\Delta\|$$ where $\|{\cdot}\|$ is some appropriate matrix norm.
A matrix would then be more symmetric than another if it has smaller asymmetry.