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

(4.202)

– 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 Vawould 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×1implying a T Va = 0, as seen previously – so Va = 0 n×1and a T Va = 0 are equivalent statements. Consequently, Va ≠ 0 n×1accounts for the inequality case in Eq. (4.196), i.e. a T Va> 0.

If Pdenotes an ( n × m ) matrix, then the ( m × m ) matrix P T VPis (real) symmetric, positive semidefinite; in fact,

(4.206)

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

(4.207)

Furthermore, the scalar

(4.208)

– where aand bdenote ( 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 × 1and 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 Vis 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 nis 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 Pis 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.

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 R 9domain; 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

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)