matrix of jacobi the transpose of this matrix is known as the matrix of jacobi of at (sometimes called the frechet derivative of at )that isf # ( :( fm fm fm xn if we define ( : (xand take the "vector/vectorderivative of with respect to xwe obtain the matrix of second-order partial derivatives of xm xm ( :( xm xm xm hessian matrix which is known as the hessian matrix of at xalso denoted as (xif these secondf xj andhenceorder partial derivatives are continuous in region around xthen xi the hessian matrix (xis symmetric finallynote that we can also define "scalar/matrixderivative of with respect to rmxn with (ij)-th entry xi xy xy : xymn xm xm and "matrix/scalarderivativex : xm xm ******** xmn example (scalar/matrix derivativelet axbwhere rmxn rm and rn since is scalarwe can write tr(ytr(xba)using the cyclic property of the trace (see theorem defining :bawe have ym [xc]ii = = = so that /xi ji orin matrix formy cabx xi ji