v = v(x) || | colspan=2| |- | u = u(x) || || || |- | u = u(x) || || || |} Scalar-by-vector identities The fundamental identities are placed above the thick black line. {|class="wikitable" style="text-align: center;" |+ Identities: scalar-by-vector ! scope="col" width="150" | Condition ! scope="col" width="200" | Expression ! scope="col" width="200" | Numerator layout,i.e. by xT; result is row vector ! scope="col" width="200" | Denominator layout,i.e. by x; result is column vector |- | 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| |- | u = u(x), v = v(x) | | in numerator layout | in denominator layout |- | u = u(x), v = v(x), A is not a function of x | | in numerator layout | in denominator layout |- | | | | , the Hessian matrix |- style="border-top: 3px solid;" | a is not a function of x || || || |- | A is not a function of x b is not a function of x || || || |- | A is not a function of x || || || |- | A is not a function of x A is symmetric || || || |- | A is not a function of x || || colspan=2| |- | A is not a function of x A is symmetric || || colspan=2| |- | || || || |- | a is not a function of x, u = u(x) | | in numerator layout | in denominator layout |- | a, b are not functions of x || || || |- | A, b, C, D, e are not functions of x || || || |- | a is not a function of x || |||| |} Vector-by-scalar identities {|class="wikitable" style="text-align: center;" |+ Identities: vector-by-scalar ! scope="col" width="150" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Numerator layout, i.e. by y,result is column vector ! scope="col" width="100" | Denominator layout, i.e. by yT,result is row vector |- | a is not a function of x || ||colspan=2| |- | a is