v2009.01.01 - Convex Optimization

542 APPENDIX A. LINEAR ALGEBRA
A.2 Semidefiniteness: domain of test
The most fundamental necessary, sufficient, and definitive test for positive
semidefiniteness of matrix A ∈ R n×n is: [177,1]
x T Ax ≥ 0 for each and every x ∈ R n such that ‖x‖ = 1 (1334)
Traditionally, authors demand evaluation over broader domain; namely,
over all x ∈ R n which is sufficient but unnecessary. Indeed, that standard
textbook requirement is far over-reaching because if x T Ax is nonnegative for
particular 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 where
superscript H denotes conjugate transpose) A.2 are admitted to the set of
positive semidefinite matrices (1337); an unnecessary prohibitive condition.
A.2.1
Symmetry versus semidefiniteness
We call (1334) the most fundamental test of positive semidefiniteness. Yet
some authors instead say, for real A and complex domain ({x∈ C n }), the
complex test x H Ax≥0 is most fundamental. That complex broadening of the
domain of test causes nonsymmetric real matrices to be excluded from the set
of positive semidefinite matrices. Yet admitting nonsymmetric real matrices
or not is a matter of preference A.3 unless that complex test is adopted, as we
shall now explain.
Any real square matrix A has a representation in terms of its symmetric
and antisymmetric parts; id est,
A = (A +AT )
2
+ (A −AT )
2
(46)
Because, for all real A , the antisymmetric part vanishes under real test,
x T(A −AT )
2
x = 0 (1335)
A.2 Hermitian symmetry is the complex analogue; the real part of a Hermitian matrix
is symmetric while its imaginary part is antisymmetric. A Hermitian matrix has real
eigenvalues and real main diagonal.
A.3 Golub & Van Loan [134,4.2.2], for example, admit nonsymmetric real matrices.