Читать книгу Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata - Страница 44

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. (4.2), where Eq. (4.4) prompts transformation to

(4.18)

in view of 0 being the neutral element for addition of scalars; Eq. (4.18) finally gives rise to

(4.19)

again at the expense of Eq. (4.2). Therefore, 0_m×n plays the role of neutral element with regard to addition of matrices, i.e. it leaves the other (m × n) matrix (to which it is added) unchanged.