v2007.09.13 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 219d 2d2g(X+ t Y ) =dt2 dt 2(X+ t Y )2 = 2Y 2 (536)which is positive semidefinite when Y is symmetric because then Y 2 = Y T Y(1257). 3.13A more appropriate matrix-valued counterpart for f is g(X)=X T Xwhich is a convex function on domain X ∈ R m×n , and strictly convexwhenever X is skinny-or-square full-rank. This matrix-valued function canbe generalized to g(X)=X T AX which is convex whenever matrix A ispositive semidefinite (p.572), and strictly convex when A is positive definiteand X is skinny-or-square full-rank (Corollary A.3.1.0.5). 3.2.3.0.5 Example. Matrix exponential.The matrix-valued function g(X) = e X : S M → S M is convex on the subspaceof circulant [117] symmetric matrices. Applying the line theorem, for all t∈Rand circulant X, Y ∈ S M , from Table D.2.7 we haved 2dt 2eX+ t Y = Y e X+ t Y Y ≽S M +0 , (XY ) T = XY (537)because all circulant matrices are commutative and, for symmetric matrices,XY = Y X ⇔ (XY ) T = XY (1275). Given symmetric argument, the matrixexponential always resides interior to the cone of positive semidefinitematrices in the symmetric matrix subspace; e A ≻0 ∀A∈ S M (1631). Thenfor any matrix Y of compatible dimension, Y T e A Y is positive semidefinite.(A.3.1.0.5)The subspace of circulant symmetric matrices contains all diagonalmatrices. The matrix exponential of any diagonal matrix e Λ exponentiateseach individual entry on the main diagonal. [182,5.3] So, changingthe function domain to the subspace of real diagonal matrices reduces thematrix exponential to a vector-valued function in an isometrically isomorphicsubspace R M ; known convex (3.1.1) from the real-valued function case[46,3.1.5].There are, of course, multifarious methods to determine functionconvexity, [46] [30] [87] each of them efficient when appropriate.3.13 By (1276) inA.3.1, changing the domain instead to all symmetric and nonsymmetricpositive semidefinite matrices, for example, will not produce a convex function.