Metric for how symmetric a matrix is

Question

Given a square NxN matrix A, what is a measure of how symmetric A is?

I can get the symmetric and antisymmetric parts of A as:

$A_{sym}=\frac{1}{2}(A+A^{T})$

and

$A_{anti}=\frac{1}{2}(A-A^{T})$

Is there some commonly used function, $F(A,A_{sym},A_{anti})$, that gives a measure of how symmetric a matrix is? E.g. something like the ratio of the determinants of $A_{sym}$ and $A_{anti}$?

Answer 1

One simple possibity:

$s \equiv (|A_{sym}|-|A_{anti}|)/(|A_{sym}|+|A_{anti}|)$

Here |·| is whatever matrix norm you choose. Then $-1\le s \ \le +1$ with the lower bound saturated for an antisymmetric matrox, upper bound saturated for a symmetric one.