# Jacobian Matrix and dot product question

I have a question on how the Jacobian matrix is used in a specific example from Wiki. I present definitions first and then a question.

The Jacobian is defined as:

$$J = \begin{bmatrix} \frac {\partial f}{\partial x_1} ... \frac {\partial f}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \Delta^T f_1 \\ ... \\ \Delta^T f_m \end{bmatrix} = \begin{bmatrix} \frac {\partial f_1}{\partial x_1} ... \frac {\partial f_1}{\partial x_n} \\ ... \\ \frac {\partial f_m}{\partial x_1} ... \frac {\partial f_m}{\partial x_n} \end{bmatrix}$$

Wiki then says: $$f(y) = f(x) + J(x) \cdot (y - x)$$ is the best linear approximation for $$f(y)$$ for points close to $$x$$.

Questions

1. Does $$J(x)$$ represent scalar multiplication of the initial point $$x$$ with the $$J$$? I.e. to get the PDs at a point.

2. If (1) above, how does one distinguish this operation from matrix multiplication?

3. For $$J(x) \cdot (y - x)$$ can the dot product, $$\cdot$$, be omitted as matrix multiplication will result in the same value? The third form of $$J$$ is a matrix and dot product $$\cdot$$ is a vector operation, so this is confusing.

• BTW, that $\Delta^T f$ should probably be $(\nabla f)^\top$. Jan 2, 2021 at 22:51

The Jacobian matrix $$J$$ contain partial derivatives $$\frac{\partial f_i}{\partial x_j}$$ which are themselves functions. The entries of $$J(x)$$ are these partial derivatives evaluated at $$x$$, i.e. $$\frac{\partial f_i}{\partial x_j}(x)$$.

• I thought $J(x)$ would be each entry of $J$ matrix multiplied by $x$. $J$ seems to represent two different objects.So $J_f(x) = J * x$?
– Nick
Jan 2, 2021 at 22:02
• @Nick No, as I said in my answer, $J(x)$ is a matrix whose $(i,j)$ entry is the real number $\frac{\partial f_i}{\partial x_j}(x)$. There is no multiplication going on in the definition of $J(x)$. Jan 2, 2021 at 22:05
• My confusion was over $J(x)$ with $x$ as supplied argument. Given what you say, a matrix of functions can all each be evaluated with a single argument $(x)$ supplied to the matrix?
– Nick
Jan 2, 2021 at 22:38
• @Nick Yes, that is how I think it should be interpreted: $J$ is a matrix of functions, and $J(x)$ is the matrix consisting of the evaluations of those functions at $x$. Jan 2, 2021 at 22:58

As has been said before, $$J(x)$$ is the Jacobian matrix at the point $$x$$, as all the functions depend on $$x$$. The dot does not below there, as we do not take the dot product of a matrix and a vector. Just delete the dot and multiply the Jacobian matrix by the (column) vector $$y-x$$.

You don't need to worry about defining Jacobian for this. This is simple Taylor's theorem:

$$f_i\left(y\right)=f_i\left(x+(y-x)\right)=f_i\left(x\right)+\sum_j \partial_j f_i\left(x\right)\,\left(y_j-x_j\right)+\mathcal{O}\left(\left|y-x\right|^2\right)$$

for all $$i$$