Читать книгу Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata - Страница 52

One interesting property of a symmetric (n × n) matrix V, defined as

(4.187)

and consistent with Eq. (4.107), entails the alternate product by vector column a, of length n, viz.

(4.188)

and vector column b, also of length n, viz.

(4.189)

the product of a^T as per Eq. (4.188) by V as per Eq. (4.187) reads

(4.190)

following Eq. (4.47) for the algorithm of multiplication of matrices, coupled with Eq. (4.105) to calculate a^T – whereas a further multiplication of a^T V by b unfolds

(4.191)

By the same token, one may write

(4.192)

based on Eqs. (4.47), (4.105), (4.187), and (4.189) – where postmultiplication by a produces

(4.193)

if subscripts i and j are exchanged – as allowed because they are dummy variables, then Eq. (4.193) becomes

(4.194)

with the aid of the interchangeability of summations (as their lower and upper bounds are unconnected), besides v_i,j ≡ v_j,i as per Eq. (4.187). One realizes that Eq. (4.194) mimics Eq. (4.191), i.e.

(4.195)

– as long as the products exist and V is symmetric.