linear systems&normalize

Question

suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear function which can be represented by a $n \times n$ matrix. Then the jacobian of $f$ is the same as the function for $f$. But I now want to normalize the first component of the output vector to be $1$. which means for the matrix of $f$, I need to divide each row by the value of the output's first component. The jacobian this time becomes much more complicated. Since I am calculating the $L_1$ matrix norm for the jacobian, is there a way that I can keep my new jacobian as simple as possible?

Answer 1

To simplify the computation, use the chain rule: the Jacobian matrix of composition is the product of Jacobian matrices. More precisely, $D(h\circ f) = (D(h)\circ f ) D(f)$. The map you describe is the composition of the original linear map (constant Jacobian) and the map $$h(y_1,\dots,y_n) = (1,y^{-1}y_2,\dots, y^{-1}y_n) ^T$$ for which the Jacobian is easy to write down: $$Dh(y_1,\dots,y_n) = \begin{pmatrix}0 & 0 & 0 & \dots & 0 \\ -y_1^{-2}y_2 & y_1^{-1} & 0 & \dots & 0 \\ -y_1^{-2}y_3 & 0 & y_1^{-1} & \dots & 0 \\ \dots & \dots & \dots &\dots & \dots \end{pmatrix}$$ Then you plug in $f$, multiply on the right by constant matrix $Df$, and try compute the norm... I don't promise that this will be easy, but that's what those computations are.