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

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matrices are square (m × m) identity matrices, containing only 1’s in the main diagonal, and denoted as I_m . A nil matrix is formed only by zeros, and is usually denoted as 0_m×n .

      When elements symmetrically placed relative to the diagonal are the same, the matrix is termed symmetric; all diagonal matrices are obviously symmetric. The sum of the elements in the main diagonal of matrix A is denoted as trace, abbreviated to tr A . When rows and columns of matrix A with generic element a_i,j are exchanged – with elements retaining their relative location within each row and each column, its transpose A^T results; it is accordingly described by [a_j,i]. Finally, the requirement for equality of two matrices is their sharing the same type (i.e. identical number of rows and identical number of columns) and the same spread (i.e. identical numbers in homologous positions).

4.1 Addition of Matrices

      Consider a generic (m × n) matrix A, defined as

(4.1)

      – or via its generic element (and thus in a more condensed form)

(4.2)

      where subscript_i refers to ith row and subscript_j refers to jth column; if another matrix, B, also of type (m × n), is defined as

(4.3)

      then A and B can be added according to the algorithm

(4.4)

      – so the sum will again be a matrix of (m × n) type.

      Addition of matrices is commutative; in fact,

(4.5)

      may be handled as

(4.6)

in view of Eq. (4.4) – where the commutative property of addition of scalars was taken advantage of; after using Eq. (4.5) backward, one gets

      (4.7)

from Eq. (4.6), and finally

(4.8)

as per Eqs. (4.2) and (4.3) – thus confirming the initial statement.

      If a third matrix C is defined as

(4.9)

      then one can write

      (4.10)

      together with Eqs. (4.2) and (4.3); based on Eq. (4.4), one has that

      (4.11)

and a further utilization of Eq. (4.4) leads to

(4.12)

      – along with the associative property borne by addition of scalars. One may repeat the above reasoning by first associating A and B, viz.

      (4.13)

at the expense of Eqs. (4.2), (4.3), and (4.9), with Eq. (4.4) allowing transformation to

      (4.14)

      supplementary use of Eq. (4.4) unfolds

(4.15)

with the aid of the associative property of addition of scalars, while elimination of the right‐hand side between Eqs. (4.12) and (4.15) gives rise to

      (4.16)

      – meaning that addition of matrices is associative.

      For every (m × n) matrix A, there is a null matrix 0_m×n such that

      (4.17)

      in agreement with Eq.

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