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

      By the same token, one may write

      (4.192)

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

(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,i as 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 V is symmetric.

      4.6.2 Positive Semidefinite Matrix

      A (real) symmetric, positive semidefinite (n × n) matrix V satisfies 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 Va represents a scalar. In the case of V being 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×1 implies 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 b denoting 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 V is, 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 λ² – 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, r₁ = α + ιβ and r₂ = α − ιβ – so Eq. (2.182) indicates that P{λ} = a₂(λ − r₁)(λ − r₂), or else P{λ} = a₂(λ − (α + ιβ))(λ − (α − ιβ)) = a₂((λ − α) − ιβ)((λ − α) + ιβ), where the said product of two conjugate binomials reduces to P{λ} = a₂((λ − α)² − ι² β²), or P{λ} = a₂((λ − α)² + β²) since −ι² = 1, with (λ − α)² + β² > 0 for being the sum of two squares; or a double root r – in which case P{λ} = a₂(x − r)² is also based on Eq. (2.182). In either case, the factor multiplying a₂ being positive will be unable to change the sign brought about by a₂. Moreover, V is positive semidefinite by hypothesis, so a^T Va is nonnegative in agreement with Eq. (4.196) – and exchange of a with any other (compatible) vector b will 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.

(4.202)