Can you help me understand the statement given below, particularly what is said when evaluating (15) and the conclusion leading to, and given by, (16)?
On a mathematical level, in view of the scalar linear phenomenological relation (10) in the neighbourhood of the equilibrium, the scalar internal power density (8) is a positive semi-definite quadratic form of the generalised scalar force vector $F$, $$\tag{12} p_S^{int}=F^TLF\ge 0$$ Since the scalar internal power density $p_S^{int}=F^TLF$ is a quadratic form, it is equal to its transpose, $$\tag{13} F^TLF=(F^TLF)^T=F^TL^TF$$ The scalar Onsager matrix $L$ is written as the sum of a symmetric matrix $S$ and an antisymmetric matrix $A$,
$$\tag{14} L = S + A \ \ \text{where} \ \ S = S^T \ \ \text{and} \ \ A =-A^T$$
In view of the decomposition (14), the quadratic form (13) is recast as, $$\tag{15} F^TSF+F^TAF=(F^TSF)^T+(F^TAF)^T=F^TS^TF+F^TA^TF=F^TSF-F^TAF$$
which shows that the antisymmetric part of the positive semi-definite quadratic form does not contribute to the scalar internal power density $p_S^{int}$, $$\tag{16} F^TAF=-F^TA^TF=0$$
Looking at the second and very last terms of (15), I think we can say, $$F^TAF=-F^TAF$$
which does not equate to what is given for (16). This is my confusion. Can you help me reconcile this? And can you further explain the claim leading to (16), specifically, "...the antisymmetric part of the positive semi-definite quadratic form does not contribute...," can you help me see why this is so?
