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First and Second Order Differentials



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First and Second Order Di erentials Jacobian and Hessian Matrices Table 1 Classi cation of functions and variables Scalar variable Vector variable Matrix variable Scalar function x X Vector function f f x f X Matrix function F F x F X Function x X f f x f X F F x F X Table 2 Di erential d d d aT dx d tr AT dX vec A T dvec X df ad df Adx df Advec X dF Ad dvec F Adx dvec F Advec X The rst identi cation Derivative Jacobian D D x aT D x vec A T Df a Df x A Df X A DF vec A DF x A DF X A table Order of D Other notation 1 1 1 n X 1 nq A X m 1 m n m nq mp 1 mp n mp nq F A Table 3 Scalar function of a vector x d x D x T T a x a dx aT T T T T x Ax x A A dx x A AT References 1 J R Magnus H Neudecker Matrix Di erential Calculus with Application in Statistics and Econometrics John Wiley 1988 UConn Library Call number QA188 M345 1988 1 Table 4 Trace of a matrix X d X D X tr AX tr AdX vec AT T T T T T T T tr XAX B tr AX B A X B dX vec B XAT BXA T tr XAXB tr AXB BXA dX vec BT XT AT AT XT BT T Table 5 Determinant of a matrix X d X D X 1 X X tr X dX X vec X 1 T T XXT 2 XXT tr XT XXT 1 dX 2 XXT vec XXT 1 X T XT X 2 XT X tr XTX 1 XT dX 2 XT X vec X XT X 1 T X2 2 X2 tr X 1dX 2 X2 vec X 1 T X X AT T B XAT BXA BT XT AT AT XT BT X X X X 1 T 2 XXT XXT 1 X 2 XT X vec X XT X 1 T 2 X2 X 1 T Table 6 Matrix functions F X dF X DF X X dX Inq T T X dX Kn q XXT dX XT X dX T 2Nn X In XT X dX T X XT dX 2Nq Iq XT Table 7 More matrix functions F X dF X DF X Conditions 1 1 1 T 1 X dX X X X non singular p X j 1 p XT p j p p j j 1 X dX X X X square p 1 2 j 1 X j 1 X Function x X f f x f X F F x F X Table 8 The second identi cation table Second Di erential Hessian matrix 2 2 d d H d2 x T Bdx H x 12 B BT d2



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