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
if subscripts i and j are exchanged – as allowed because they are dummy variables, then Eq. (4.193) becomes
with the aid of the interchangeability of summations (as their lower and upper bounds are unconnected), besides vi,j ≡ vj,i as per Eq. (4.187). One realizes that Eq. (4.194) mimics Eq. (4.191), i.e.
– 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
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 aT 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
because Va =0n×1 implies aT 0n×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
while
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
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 aT Va is nonnegative in agreement with Eq. (4.196) – and exchange of a with any other (compatible) vector b will also lead to bT Vb ≥ 0, as expected. No distinct real roots can exist for P{λ}, so its discriminant binomial, ΔP, must be negative or nil, i.e.