F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

(4.71)

upon retrieval of Eqs. (4.2), (4.3), and (4.50)with p ≡ n ; in view of Eq. (4.4), it becomes possible to redo Eq. (4.71)to

(4.72)

whereas application of Eq. (4.47)allows further transformation to

(4.73)

In view of the distributive property of multiplication of scalars, Eq. (4.73)turns to

(4.74)

where splitting of the summation meanwhile took place – with the aid of the associative property of addition of scalars; the algorithm labeled as Eq. (4.47)may again be retrieved, viz.

(4.75)

along with Eq. (4.4)– which is the same to say

(4.76)

again at the expense of Eqs. (4.2), (4.3), and (4.50). By the same token,

(4.77)

– in view of the definitions of A, B, and C, via their generic elements conveyed by Eqs. (4.2), (4.46), and (4.50)with p ≡ n ; Eq. (4.4)supports transformation of Eq. (4.77)to

(4.78)

while Eq. (4.47)permits conversion to

(4.79)

Based on the distributive property of multiplication of scalars, Eq. (4.79)will give rise to

(4.80)

together with replacement of the summation of the sum by the sum of corresponding summations; Eqs. (4.4)and (4.47)may finally be called upon to write

(4.81)

or else

(4.82)

again due to Eqs. (4.2), (4.46), and (4.50). Therefore, both the pre‐ and the postmultiplication of matrices are distributive.

In the case of an ( n × n ) matrix, one may proceed to sequential multiplication k times by itself – usually labeled as

(4.83)

the outcome is still an ( n × n ) square matrix – while

(4.84)

is usually set by convention. In view of the unique features of an identity matrix outlined in Eqs. (4.61)and (4.64), one realizes that

(4.85)

– because multiplying an ( n × n ) identity matrix by itself leaves any of them unchanged.

If two matrices are partitioned in blocks, multiplication is still to follow the algorithm conveyed by Eq. (4.47)– as long as elements are replaced by germane submatrices; however, the underlying rules of compatibility between columns and rows of factor blocks are to be satisfied by all products. Consider, in this regard, the most common (and simplest) case of a (2 × 2) block matrix, viz.

(4.86)

– where ( m × n ) matrix Awas partitioned through ( m 1× n 1) matrix A 1,1, ( m 1× n 2) matrix A 1,2, ( m 2× n 1) matrix A 2,1, and ( m 2× n 2) matrix A 2,2as constitutive blocks, with m 1 + m 2 = m and n 1 + n 2 = n ; coupled with

(4.87)

– with ( p × q ) matrix Bpartitioned as ( p 1× q 1) matrix B 1,1, ( p 1× q 2) matrix B 1,2, ( p 2× q 1) matrix B 2,1, and ( p 2× q 2) matrix B 2,2as constitutive blocks – as well as p 1 + p 2 = p and q 1 + q 2 = q . The product ABis possible if n = p , besides the number of columns of the blocks of Acoinciding with the number of rows of the corresponding blocks of B. In fact, the said product will look like

(4.88)

following the regular algorithm of multiplication of matrices – as long as n 1 = p 1to allow existence of A 1,1 B 1,1, A 1,1 B 1,2, A 2,1 B 1,1, and A 2,1 B 1,2, as well as n 2 = p 2to allow calculation of A 1,2 B 2,1, A 1,2 B 2,2, A 2,2 B 2,1, and A 2,2 B 2,2. This approach is particularly advantageous when one (or more) of the foregoing blocks is either an identity or a null matrix – since the matrix elements in Eq. (4.88)would yield much simpler expressions, involving only a portion of Aor Brather than their whole set of elements. Such a concept may logically be extended to any partition as factor matrices – provided that it is mathematically feasible per se , and the associated block matrices are compatible with regard to multiplication.