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