href="#ulink_8163488a-3dbf-5a62-abf2-636e455a39e7">Eq. (4.166).
4.6 Combined Features
Among the many possible combinations of matrix operations, two of them hold a particular relevance. The first pertains to a symmetric matrix, with regard to pre‐ and postmultiplication by a vector or its transpose – while the other encompasses a related property, which eventually supports definition of a positive semidefinite matrix.
4.6.1 Symmetric Matrix
One interesting property of a symmetric (n × n) matrix V, defined as
and consistent with Eq. (4.107), entails the alternate product by vector column a, of length n, viz.
and vector column b, also of length n, viz.
the product of aT as per Eq. (4.188) by V as 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 aT – whereas a further multiplication of aT V by b unfolds
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
