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

where 4 was meanwhile factored out and the associative property of the multiplication of scalars duly utilized; if a^T Va = 0 as stated in Eq. (4.197), then Eq. (4.202) reduces to

      (4.203)

      that implies

(4.204)

because any other value of b^T Va would make the left‐hand side positive for it being a square. Equation (4.204) implies, in turn,

      (4.205)

      since no restriction was imposed on b; therefore, a^T Va = 0 implies Va =0_n×1, besides Va =0_n×1 implying a^T Va = 0, as seen previously – so Va =0_n×1 and a^T Va = 0 are equivalent statements. Consequently, Va ≠0_n×1 accounts for the inequality case in Eq. (4.196), i.e. a^T Va > 0.

      If P denotes an (n × m) matrix, then the (m × m) matrix P^T VP is (real) symmetric, positive semidefinite; in fact,

      (4.206)

as per Eqs. (4.110) and (4.120) – so P^T VP abides to the symmetry requirement because, by hypothesis, V^T = V, i.e.

(4.207)

      Furthermore, the scalar

      (4.208)

      – where a and b denote (m × 1) and (n × 1) column vectors, respectively, and Eqs. (4.57) and (4.120) were taken advantage of, will be positive if b ≡ Pa ≠ 0_{n × 1} and nil if b ≡ Pa = 0_{n × 1}, in agreement with the foregoing derivation, coupled with Eqs. (4.196) and (4.197); this is so because V is positive semidefinite by hypothesis. In other words,

(4.209)

– which, together with Eq. (4.207), confirms the full original statement on P^T VP, i.e. a symmetric and positive definite matrix.

      In the particular case of V = I_n, Eq. (4.196) degenerates to

      (4.210)

at the expense of the n‐dimensional analogue of Eq. (3.8) – with the aid of Eq. (4.64) and explicitation of (n × 1) matrix a; hence, I_n is positive semidefinite as per Eq. (4.196). One may therefore resort directly to Eq. (4.209) to write

(4.211)

encompassing (m × 1) matrix a; and Eq. (4.211) may be algebraically simplified to

(4.212)

in view of Eq. (4.64). Inspection of Eq. (4.212) confirms that P^T P is necessarily a positive semidefinite matrix, while

      (4.213)

      as per Eq. (4.120), or else

      (4.214)

      in view of Eq. (4.110) – thus confirming its symmetry.

5 Tensor Operations

      A tensor, τ, consists on the extrapolation of a vector to a three‐dimensional space, based on each of its three directional components – so it has no geometrical representation, being defined solely by its components in a Cartesian R⁹ domain; its notation resorts normally to matrix form, viz.

(5.1)

      – so it can be formed by juxtaposition of three vectors of the type labeled as Eq. (3.6), or else

(5.2)

      using Eq. (3.1) as template. The unit tensors, φ_i,j (i = x,y,z; j = x,y,z), are, in turn, defined as arrays of nine components (all of which are zero but one, equal to unity), according to