v2007.09.17 - Convex Optimization

A.2. SEMIDEFINITENESS: DOMAIN OF TEST 485A.2 Semidefiniteness: domain of testThe most fundamental necessary, sufficient, and definitive test for positivesemidefiniteness of matrix A ∈ R n×n is: [151,1]x T Ax ≥ 0 for each and every x ∈ R n such that ‖x‖ = 1 (1234)Traditionally, authors demand evaluation over broader domain; namely,over all x ∈ R n which is sufficient but unnecessary. Indeed, that standardtextbook requirement is far over-reaching because if x T Ax is nonnegative forparticular x = x p , then it is nonnegative for any αx p where α∈ R . Thus,only normalized x in R n need be evaluated.Many authors add the further requirement that the domain be complex;the broadest domain. By so doing, only Hermitian matrices (A H = A wheresuperscript H denotes conjugate transpose) A.2 are admitted to the set ofpositive semidefinite matrices (1237); an unnecessary prohibitive condition.A.2.1Symmetry versus semidefinitenessWe call (1234) the most fundamental test of positive semidefiniteness. Yetsome authors instead say, for real A and complex domain (x∈ C n ), thecomplex test x H Ax≥0 is most fundamental. That complex broadening of thedomain of test causes nonsymmetric real matrices to be excluded from the setof positive semidefinite matrices. Yet admitting nonsymmetric real matricesor not is a matter of preference A.3 unless that complex test is adopted, as weshall now explain.Any real square matrix A has a representation in terms of its symmetricand antisymmetric parts; id est,A = (A +AT )2+ (A −AT )2(44)Because, for all real A , the antisymmetric part vanishes under real test,x T (A −AT )2x = 0 (1235)A.2 Hermitian symmetry is the complex analogue; the real part of a Hermitian matrixis symmetric while its imaginary part is antisymmetric. A Hermitian matrix has realeigenvalues and real main diagonal.A.3 Golub & Van Loan [110,4.2.2], for example, admit nonsymmetric real matrices.