v2007.09.13 - Convex Optimization

566 APPENDIX D. MATRIX CALCULUSIn the case g(X) : R K →R has vector argument, they further simplify:→Ydg(X) = ∇g(X) T Y (1603)→Ydg 2 (X) = Y T ∇ 2 g(X)Y (1604)and so on.→Ydg 3 (X) = ∇ X(Y T ∇ 2 g(X)Y ) TY (1605)D.1.7Taylor seriesSeries expansions of the differentiable matrix-valued function g(X) , ofmatrix argument, were given earlier in (1577) and (1598). Assuming g(X)has continuous first-, second-, and third-order gradients over the open setdomg , then for X ∈ dom g and any Y ∈ R K×L the complete Taylor serieson some open interval of µ∈R is expressedg(X+µY ) = g(X) + µ dg(X) →Y+ 1 →Yµ2dg 2 (X) + 1 →Yµ3dg 3 (X) + o(µ 4 ) (1606)2! 3!or on some open interval of ‖Y ‖g(Y ) = g(X) +→Y −X→Y −X→Y −Xdg(X) + 1 dg 2 (X) + 1 dg 3 (X) + o(‖Y ‖ 4 ) (1607)2! 3!which are third-order expansions about X . The mean value theorem fromcalculus is what insures finite order of the series. [160] [30,1.1] [29, App.A.5][147,0.4]D.1.7.0.1 Exercise. log det. (confer [46, p.644])Find the first two terms of the Taylor series expansion (1607) for log detX .