not a function of x,u = u(x) || | colspan=2| |- | A is not a function of x,u = u(x) || || || |- | u = u(x) || | colspan=2| |- | u = u(x), v = v(x) || | colspan=2| |- | u = u(x), v = v(x) || || || |- | rowspan=2|u = u(x) || rowspan=2| || || |- |colspan=2|Assumes consistent matrix layout; see below. |- | rowspan=2|u = u(x) || rowspan=2| || || |- |colspan=2|Assumes consistent matrix layout; see below. |- | U = U(x), v = v(x) || || || |} NOTE: The formulas involving the vector-by-vector derivatives and (whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives. Scalar-by-matrix identities Note that exact equivalents of the scalar product rule and chain rule do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the trace function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.: For example, to compute Therefore, (numerator layout) (For the last step, see the Conversion from differential to derivative form section.) {|class="wikitable" style="text-align: center;" |+ Identities: scalar-by-matrix ! scope="col" width="175" | Condition ! scope="col" width="10" | Expression ! scope="col" width="100" | Numerator layout, i.e. by XT ! scope="col" width="100" | Denominator layout, i.e. by X |- | a is not a function of X || | || |- | a is not a function of X, u = u(X) || | colspan=2| |- | u = u(X), v = v(X) || | colspan=2| |- | u = u(X), v = v(X) || | colspan=2| |- | u = u(X) || | colspan=2| |- | u = u(X) || | colspan=2| |- | rowspan=2|U = U(X) || rowspan=2| || || |- | colspan=2|Both forms assume numerator layout for i.e. mixed layout if denominator layout for X is being used. |- style="border-top: 3px solid;" | a and b are not functions of X || ||| |- | a and b are not functions of X || ||| |- | a, b and C are not functions of X || ||| |- | a, b and C are not functions