it follows immediately that if is symmetricthat isa athen (xax ax and (xax to prove ( )first observe that (xxax ni= nj= ai xi which is quadratic form in xis real-valuedwith xx aik xi ak ai xi xk xk = = = = the first term on the right-hand side is equal to the -th element of axwhereas the second term equals the -th element of xaor equivalently the -th element of ax taylor expansion the matrix of jacobi and the hessian matrix feature prominently in multidimensional taylor expansions theorem multidimensional taylor expansions let be an open subset of rn and let if is continuously twice differentiable function with jacobian matrix (xand hessian matrix ( )then for every we have the following firstand second-order taylor expansionsf (xf (aj ( ( ao(kx ak ( and (xf (aj ( ( ( ) ( ( ao(kx ak ( as kx ak by dropping the remainder termsone obtains the corresponding taylor approximations the result is essentially saying that smooth enough function behaves locally (in the neighborhood of point xlike linear and quadratic function thusthe gradient or hessian of an approximating linear or quadratic function is basic building block of many approximation and optimization algorithms remark (version without remainder termsan alternative version of taylor' theorem states that there exists an that lies on the line segment between and such that ( and ( hold without remainder termswith (ain ( replaced by ( and (ain ( replaced by ( composition chain rule consider the functions rk rm and rm rn the function gf ( )is called the composition of and written as and is function from rk to rn suppose (xand ( )as in figure let (xand (ybe the (frechetderivatives of (at xand (at )respectively we may think of (xas the matrix that describes