(from MathFlow) 设 $A=(a_{ij})$, 且定义 $$ex _A f(A)=sex{cfrac{p f}{p a_{ij}}}. eex$$ 试证: (1) $ _A r (AB)=B^t$; (2) $ _A r(ABA^tC)=CAB+C^tAB^t$.
证明: (1) $$eex ea _A r (AB) &=sex{cfrac{p }{p a_{ij}}sum_{m,n}a_{mn}b_{nm}}\ &=sex{sum_{m,n} delta_{mi}delta_{nj}b_{nm}}\ &=sex{b_{ji}}\ &=B^t. eea eeex$$ (2) $$eex ea _A r (ABA^tC) &=sex{cfrac{p }{p a_{ij}} sum_{m,n,p,q} a_{mn}b_{np}a_{qp}c_{qm} }\ &=sex{ sum_{m,n,p,q} delta_{mi}delta_{nj}b_{np}a_{qp}c_{qm} +sum_{m,n,p,q} a_{mn}b_{np}delta_{qi}delta_{pj}c_{qm} }\ &=sex{ sum_{p,q} b_{jp}a_{qp}c_{qi} +sum_{m,n} a_{mn}b_{nj}c_{im} }\ &=sex{ sum_{p,q} c_{qi}a_{qp}b_{jp} +sum_{m,n} c_{im}a_{mn}b_{nj} }\ &=C^tAB^t+CAB. eea eeex$$