Is jacobian matrix of a function is the direction of descent?

Question

Since the gradient of a function shows the steepest descent direction, can I assume that the jacobian matrix of a function implies the direction of steepest descent in case of multiobjective optimization? In that case if I have the Jacobian of more than one function and if I want to calculate a common descent direction for both the functions, then should I add both the jacobian matrices and consider it as the direction of steepest descent for both the functions?

Answer 1

The gradient of $$ \|f\|=\sqrt{f_1^2+...+f_n^2} $$ is the vector $$ \nabla\|f\|=\frac{\sum_{k=1}^nf_k\nabla f_k}{\|f\|} $$ and gives the direction of the steepest ascent of $\|f\|\,.$ Since the vector $\nabla f_k=(\partial_1 f_k,...,\partial_mf_k)$ is the $k$-th row of the Jacobian matrix $J$ of $f$ it turns out that the vector $\nabla\|f\|$ is the matrix vector product $$ \nabla\|f\|=\frac{J^\top f}{\|f\|}\,. $$