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

where the distributive property of scalars allows transformation to

(3.149)

algebraic rearrangement at the expense of the commutative and associative properties of multiplication of scalars leads then to

(3.150)

so Eqs. (3.22)and (3.51)may be invoked to write

(3.151)

that retrieves Eq. (3.128)after applying Eq. (3.143)twice – thus confirming validity of Eq. (3.128), through an independent derivation path.

Finally, it is worth mentioning that the volume, V , of a parallelepiped defined by vectors u, v, and wcan be calculated as the area of the parallelogram that constitutes its base, defined by uand vand represented by vector (‖ u‖‖ v‖ sin {∠ u, v}) nas per Eq. (3.112)and (3.113), multiplied by its height – i.e. the projection of wupon n, and calculated as ‖ w‖ cos {∠ w, n} as per Eq. (3.54). On the one hand, (‖ u‖‖ v‖ sin {∠ u, v}) nis, by definition, equal to u× vas per Eq. (3.111)– so ‖ u‖‖ v‖ sin {∠ u, v} coincides with ‖ u× v‖ because ‖ n‖ = 1; on the other hand, the length of the projection of wonto nreads ‖ w‖ cos {∠ w, n}, where cos{∠ w, n} = cos {∠ w, u× v} since u× vhas the direction of n. The product of ‖ u× v‖ by ‖ w‖ cos {∠ w, u× v} is but the scalar product of u× vby was per Eq. (3.54)– so one finally concludes that

(3.152)

with a scalar quantity being now at stake.

Matrix is a nuclear concept in linear algebra; arrays of (real) numbers possess a long history associated to solution of linear equations – and records indicate that Italian mathematician Girolamo Cardano first brought a related method from China to Europe in 1545, using his book Ars Magna as vehicle. The first explicit mention to a matrix appeared in 1851 by the hands of James J. Sylvester, an English mathematician – although in the context of determinants. Since he was interested in the determinant formed from a rectangular array of numbers and not in the array itself, he coined the word matrix from the Latin mater meaning womb (i.e. the place from which something else originates); it remained up to his collaborator Arthur Cayley to ascribe the modern sense to the concept of matrix.

Being an array of numbers, arranged as m rows × n columns, and enclosed by a set of square parenthesis, [ ai,j ] with i =1,2,…, m and j =1,2,…, n , a real matrix actually originates from R m×n. It is termed rectangular when m ≠ n , and square when m = n ; and reduces to a row vector when m = 1, or a column vector when n = 1. The main diagonal is formed by elements of the type a i,i; if all entries below the main diagonal are zero, the matrix is said to be upper triangular and lower triangular when all entries above the main diagonal are nil. A diagonal matrix is both upper and lower triangular, i.e. all elements off the main diagonal are zero; if all elements in the diagonal are, in turn, equal to each other, then a scalar matrix arises. The most important scalar 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 Ais denoted as trace, abbreviated to tr A . When rows and columns of matrix Awith generic element a i,jare exchanged – with elements retaining their relative location within each row and each column, its transpose A Tresults; 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).

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 irefers to i th row and subscript jrefers to j th column; if another matrix, B, also of type ( m × n ), is defined as

(4.3)

then Aand Bcan 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