v2007.09.17 - Convex Optimization

486 APPENDIX A. LINEAR ALGEBRAonly the symmetric part of A , (A +A T )/2, has a role determining positivesemidefiniteness. Hence the oft-made presumption that only symmetricmatrices may be positive semidefinite is, of course, erroneous under (1234).Because eigenvalue-signs of a symmetric matrix translate unequivocally toits semidefiniteness, the eigenvalues that determine semidefiniteness arealways those of the symmetrized matrix. (A.3) For that reason, andbecause symmetric (or Hermitian) matrices must have real eigenvalues,the convention adopted in the literature is that semidefinite matrices aresynonymous with symmetric semidefinite matrices. Certainly misleadingunder (1234), that presumption is typically bolstered with compellingexamples from the physical sciences where symmetric matrices occur withinthe mathematical exposition of natural phenomena. A.4 [96,52]Perhaps a better explanation of this pervasive presumption of symmetrycomes from Horn & Johnson [150,7.1] whose perspective A.5 is the complexmatrix, thus necessitating the complex domain of test throughout theirtreatise. 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 thedefinition of positive definiteness. It is customary, however.Their comment is best explained by noting, the real part of x H Ax comesfrom the Hermitian part (A +A H )/2 of A ;rather,Re(x H Ax) = x H A +AH2x (1236)x H Ax ∈ R ⇔ A H = A (1237)because the imaginary part of x H Ax comes from the anti-Hermitian part(A −A H )/2 ;Im(x H Ax) = x H A −AH x2(1238)that vanishes for nonzero x if and only if A = A H . So the Hermitiansymmetry assumption is unnecessary, according to Horn & Johnson, notA.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 positivesemidefinite cone, for example, is not self-dual (2.13.5) in the ambient space of Hermitianmatrices. [145,II]