Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier MalcataЧитать онлайн книгу.
id="ulink_3b12eb31-2e51-5f28-9882-ce0689cdaadd">(4.177)
Insertion of Eq. (4.177) transforms Eq. (4.168) to
(4.178)
where B1,1 may be factored out as
(4.179)
in agreement with Eq. (4.76); isolation of B1,1 then becomes possible via premultiplication of both sides by ( A1,1 − A1,2
once again with the aid of Eqs. (4.61) and (4.124). One may likewise combine Eqs. (4.171) and (4.174) to get
(4.181)
with the aid of Eq. (4.24), where factoring out of B2,2 yields
again at the expense of Eq. (4.76), besides Eq. (4.8); after premultiplying both sides by ( A2,2 − A2,1
from Eq. (4.182). In view of Eq. (4.183), one may transform Eq. (4.174) as
(4.184)
whereas
results from combination of Eqs. (4.177) and (4.180); therefore, Eqs. (4.180) and (4.183)–(4.185) support reformulation of Eq. (4.87) finally to
(4.186)
since B = A−1 as per Eqs. (4.124) and (4.166). A similar conclusion on the form of A−1 can, as expected from the double equality in Eq. (4.124), be drawn if AB is replaced by BA in Eq. (4.166).
4.6 Combined Features
Among the many possible combinations of matrix operations, two of them hold a particular relevance. The first pertains to a symmetric matrix, with regard to pre‐ and postmultiplication by a vector or its transpose – while the other encompasses a related property, which eventually supports definition of a positive semidefinite matrix.
4.6.1 Symmetric Matrix
One interesting property of a symmetric (n × n) matrix V, defined as
and consistent with Eq. (4.107), entails the alternate product by vector column a, of length n, viz.
and vector column b, also of length n, viz.
the product of aT 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 aT – whereas a further multiplication of aT V by b unfolds