Is there any sense in zero-padding a matrix to make it $n\times n$ and find its eigenvalues?

I am debuging my Kalman filter and the Jacobian matrix of partial derivatives of h(measurement function) with respect to x(state) is not n×n, it is 13×16.

$\displaystyle \quad\ \bf H_{[i,j]}$ = $\bf \frac{\partial h_{[i]}}{\partial x_{[j]}}(\tilde x_k,0)$

I would like to extract as much information as possible and one the things I thought of was eigenvalues, but it is not square. I wonder if I can zeropadd and make it square.

Is there any sense?

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That depends on what you are doing (I have no idea about your application). But a priori filling with zeros does imo make sense. A $n\times m$ matrix can be understood as a linear map $\mathbb R^m\to \mathbb R^n$. Here you have $n=13<m=16$. You have two options now which eventually should give the same result.
First you realise that you can perform row and column operations to reduce your matrix to a non-trivial $13\times 13$ matrix and a $13\times 3$ block of zeros. This means that you split the bigger $\mathbb R^m$ into two subspaces, one of dimension $3$ which is mapped to $0$ and one of dimension $13$ which still maps surjectively on the image of the original linear map. All information is then concentrated in the $13\times 13$ block and this should give you some information.
Or you fill the matrix with zeros to get a $16\times 16$ matrix. This corresponds to changing your target space where the additional dimensions are not hit by your map. The image will then still live in a $13$-dimensional subspace of $\mathbb R^{16}$. But diagonalising will give you a nontrivial $13\times 13$ block in the $16\times 16$ matrix again. This will look like the $13\times 13$ block as above.