Derivatives, example 1

Let now $\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}$, where $\boldsymbol{A}$ is an $m\times n$ matrix and the matrix does not depend on $\boldsymbol{x}$. If we write out the vector $\boldsymbol{y}$ compoment by component we have

$$y_i = \sum_{j=0}^{n-1}a_{ij}x_j,$$

with $\forall i=0,1,2,\dots,m-1$. The individual matrix elements of $\boldsymbol{A}$ are given by the symbol $a_{ij}$. It follows that the partial derivatives of $y_i$ with respect to $x_k$

$$\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1.$$

From this we have, using the definition of the Jacobian

$$\frac{\partial \boldsymbol{y} }{\partial \boldsymbol{x}}= \boldsymbol{A}.$$

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