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

One interesting property of a symmetric ( n × n ) matrix V, defined as

(4.187)

and consistent with Eq. (4.107), entails the alternate product by vector column a, of length n , viz.

(4.188)

and vector column b, also of length n , viz.

(4.189)

the product of a Tas per Eq. (4.188)by Vas per Eq. (4.187)reads

(4.190)

following Eq. (4.47)for the algorithm of multiplication of matrices, coupled with Eq. (4.105)to calculate a T– whereas a further multiplication of a T Vby bunfolds

(4.191)

By the same token, one may write

(4.192)

based on Eqs. (4.47), (4.105), (4.187), and (4.189)– where postmultiplication by aproduces

(4.193)

if subscripts i and j are exchanged – as allowed because they are dummy variables, then Eq. (4.193)becomes

(4.194)

with the aid of the interchangeability of summations (as their lower and upper bounds are unconnected), besides v i,j≡ v j,ias per Eq. (4.187). One realizes that Eq. (4.194)mimics Eq. (4.191), i.e.

(4.195)

– as long as the products exist and Vis symmetric.

A (real) symmetric, positive semidefinite ( n × n ) matrix Vsatisfies the condition

(4.196)

for any real ( n × 1) vector a– irrespective of size or type of V, and of magnitude or sign of its elements, as long as the said product can be calculated; it should be emphasized that a T Varepresents a scalar. In the case of Vbeing symmetric – and recalling the multiplication of any significant vector by a null matrix as per Eq. (4.67), one readily finds that

(4.197)

because Va = 0 n×1implies a T 0 n×1 = 0 in general. To show the converse, one may resort to any form of column vector en lieu of a, namely, that obtained from addition of λb to a– with bdenoting a vector of appropriate dimensions and λ denoting a scalar; a quadratic polynomial P { λ } may accordingly be defined as

(4.198)

while

(4.199)

in view of Eq. (4.196)– with no restriction imposed upon column vector a, or a + λb , for that matter. Algebraic expansion of Eq. (4.198)leads to

(4.200)

at the expense of Eqs. (4.24), (4.34), (4.76), (4.82), (4.114), and (4.123); as Vis, by hypothesis, symmetric, one may resort to Eq. (4.195)to transform Eq. (4.200)to

(4.201)

also with the aid of the commutativity and associativity of addition of scalars. Since P { λ } cannot change sign as per Eq. (4.199)when λ spans the whole real domain, then its sign should remain that of the coefficient of λ 2– as will be seen later, when dealing with the roots of a quadratic equation. In fact, absence of real distinct roots (as entertained by P { λ } ≥ 0 implies one of two possibilities: either a set of conjugate complex roots, say, r 1 = α + ιβ and r 2 = α − ιβ – so Eq. (2.182)indicates that P { λ } = a 2( λ − r 1)( λ − r 2), or else P { λ } = a 2( λ − ( α + ιβ ))( λ − ( α − ιβ )) = a 2(( λ − α ) − ιβ )(( λ − α ) + ιβ ), where the said product of two conjugate binomials reduces to P { λ } = a 2(( λ − α ) 2 − ι 2 β 2), or P { λ } = a 2(( λ − α ) 2 + β 2) since −ι 2 = 1, with ( λ − α ) 2 + β 2 > 0 for being the sum of two squares; or a double root r – in which case P { λ } = a 2( x − r ) 2is also based on Eq. (2.182). In either case, the factor multiplying a 2being positive will be unable to change the sign brought about by a 2. Moreover, Vis positive semidefinite by hypothesis, so a T Vais nonnegative in agreement with Eq. (4.196)– and exchange of awith any other (compatible) vector bwill also lead to b T Vb ≥ 0, as expected. No distinct real roots can exist for P { λ }, so its discriminant binomial, Δ P, must be negative or nil, i.e.