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

(4.95), so one readily concludes that

      (4.99)

      – whereas combination with Eq. (4.57) leads finally to

(4.100)

      Remember that BC is an (n × m) matrix and A is an (m × n) matrix, so ( BC) A = BCA is an (n × n) matrix – and thus distinct from the (m × m) matrix ( AB) C = ABC as outlined above, due to the product of matrices not being commutative; nevertheless, the traces of BCA and ABC are the same. A similar derivation would prove that

      (4.101)

where (p × n) matrix CA is compatible with (n × p) matrix B, thus yielding a (p × p) matrix CAB that possesses a trace for being square – and equal to that of BCA, despite CAB being distinct from (n × n) matrix BCA . Note that the next similar move of swapping the first factor (i.e. C) to the last position without modifying the sequence of the other two (i.e. AB) would transform tr{ CAB } to tr{ ABC } again – so tr{ CAB } = tr{ ABC } would close the cycle with Eqs. (4.100) and (4.101).

      A particular situation covered by Eq. (4.100) pertains to an (n × m) matrix A and an (m × n) matrix C, together with I_m playing the role of matrix B; the products ABC and BCA in Eq. (4.100) look like

      (4.102)

      which degenerates to

      (4.103)

      at the expense of Eqs. (4.61) and (4.64). Therefore, the trace of the product of two matrices remains unchanged when the said matrices switch positions (should that be compatible with multiplication).

4.4 Transposal of Matrices

      Recall again matrix A, as defined by Eqs. (4.1) and (4.2); if generic element a_i,j, initially located in the ith row and jth column, were swapped with element a_j,i, initially located in the jth row and ith column, then a transpose matrix would result – given by

(4.104)

in more condensed form, Eq. (4.104) reads

(4.105)

      so A^T will be an (n × m) matrix. The order of a square matrix is obviously not changed upon transposal – neither do the diagonal elements (characterized by i = j), irrespective of its order; hence, one finds that

      (4.106)

      for a square A . Furthermore, if the elements symmetrically located with regard to the main diagonal are identical, then

(4.107)

      a matrix bearing this property is termed symmetric – a concept distinct from that conveyed by Eq. (4.44) that involves two matrices. A particular case of the above statement is the identity matrix – since a_i,j≠i = 0 = a_j,i; hence,

(4.108)

Application of Eq. (4.105) twice sequentially supports

      (4.109)

      which may be condensed to

(4.110)

      therefore, the inverse of transposal coincides with transposal itself, as composition of the two leaves the original matrix unchanged.

      When transposal is combined with addition of matrices, one obtains

      (4.111)

      from Eqs. (4.2) and (4.3), after direct combination with Eq. (4.105); whereas the algorithm conveyed by Eq. (4.4) gives rise to

(4.112)

A further application of Eq. (4.105) transforms Eq. (4.112) to

(4.113)

      which is equivalent to

(4.114)

      given