v2009.01.01 - Convex Optimization

A.2. SEMIDEFINITENESS: DOMAIN OF TEST 543
only the symmetric part of A , (A +A T )/2, has a role determining positive
semidefiniteness. Hence the oft-made presumption that only symmetric
matrices may be positive semidefinite is, of course, erroneous under (1334).
Because eigenvalue-signs of a symmetric matrix translate unequivocally to
its semidefiniteness, the eigenvalues that determine semidefiniteness are
always those of the symmetrized matrix. (A.3) For that reason, and
because symmetric (or Hermitian) matrices must have real eigenvalues,
the convention adopted in the literature is that semidefinite matrices are
synonymous with symmetric semidefinite matrices. Certainly misleading
under (1334), that presumption is typically bolstered with compelling
examples from the physical sciences where symmetric matrices occur within
the mathematical exposition of natural phenomena. A.4 [118,52]
Perhaps a better explanation of this pervasive presumption of symmetry
comes from Horn & Johnson [176,7.1] whose perspective A.5 is the complex
matrix, thus necessitating the complex domain of test throughout their
treatise. They explain, if A∈C n×n
...and if x H Ax is real for all x ∈ C n , then A is Hermitian.
Thus, the assumption that A is Hermitian is not necessary in the
definition of positive definiteness. It is customary, however.
Their comment is best explained by noting, the real part of x H Ax comes
from the Hermitian part (A +A H )/2 of A ;
rather,
Re(x H Ax) = x
HA +AH
x (1336)
2
x H Ax ∈ R ⇔ A H = A (1337)
because the imaginary part of x H Ax comes from the anti-Hermitian part
(A −A H )/2 ;
Im(x H Ax) = x
HA −AH
x (1338)
2
that vanishes for nonzero x if and only if A = A H . So the Hermitian
symmetry assumption is unnecessary, according to Horn & Johnson, not
A.4 Symmetric matrices are certainly pervasive in the our chosen subject as well.
A.5 A totally complex perspective is not necessarily more advantageous. The positive
semidefinite cone, for example, is not self-dual (2.13.5) in the ambient space of Hermitian
matrices. [170,II]