1
$\begingroup$

First of all, excuse the stupid question. Unfortunately, I couldn't find something similar already asked.

Consider the matrix

$$ A(r_1, r_2) = \begin{pmatrix} r_1^2 & 3 \\ 2 & r_2 \end{pmatrix}. $$

I'd like to know how I can get the Jacobian of this matrix. I'd suggest it's something like:

$$ \begin{align*} \frac{\partial A(r_1, r_2)}{\partial r_1} &= \begin{pmatrix} 2r_1 & 0 \\ 0 & 0 \end{pmatrix} \\ \frac{\partial A(r_1, r_2)}{\partial r_2} &= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \end{align*} $$

But what mathematical object is the Jacobian of the Matrix $A$? Is it a matrix with three dimensions? Any help or hints for useful links are really appreciated.

$\endgroup$
1
1
$\begingroup$

The problem is that you do not define your "space" of matrices. The easiest way to think of $2\times 2$ matrices is as $\mathbb{R}^4$ vectors. Then, the Jacobian of $A:\mathbb{R}^2\to\mathbb{R}^4$ is a $4\times 2$ matrix: $$ J_A=\begin{pmatrix} 2r_1 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 1\\ \end{pmatrix} $$

If you want to work on these derivatives as matrices, you can think of operations such as the reshape of Matlab, but it is not clear e.g. how the chain rule would work then.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.