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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 $A_{anti}$?

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  • $\begingroup$ I would just use the norm of the antisymmetric part. The symmetric matrices are a subspace of all matrices and this norm describes the distance from $A$ to the nearest symmetric matrix, which is its symmetric part. $\endgroup$ Commented Dec 7, 2016 at 23:38

1 Answer 1

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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.

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  • $\begingroup$ Yeah seems reasonable. Could also shift and scale so is a scalar on [0,1]: .5*(s+1) $\endgroup$ Commented Dec 7, 2016 at 23:38
  • $\begingroup$ @Oscar Lanzi, can you suggest some norm(s) that would be appropriate in this context? $\endgroup$
    – Reshad
    Commented Aug 28, 2020 at 5:16
  • $\begingroup$ Any matrix norm could work, so it's up to you really. $\endgroup$ Commented Aug 28, 2020 at 9:37

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