Some sort of rank-aware determinant? Would it be useful?

Question

Assume $\bf A$ has a Jordan form:

$${\bf A = SJS}^{-1}$$

If any diagonal value of $\bf J$ is $0$ then $\bf A$ does not have full rank and therefore $\det({\bf A}) = 0$.

Claim: We can define a binary selection matrix $\bf C$ so that we get something like an econ version for Jordan-decomposition similar to that of SVD:

$${\bf A = SC}^T{\bf CJC}^T{\bf CS}^{-1}$$

1. ${\bf SC}^T$ - has $\text{rank}({\bf A})$ columns.
2. ${\bf CJC}^T$ - matrix of all non-zero eigenvalues and 1s in the non-0 Jordan blocks.
3. ${\bf CS}^{-1}$ - has $\text{rank}({\bf A})$ rows.

Now would there be any use for calculating $\det({{\bf CJC}^T})$?

I'm just brainstorming. Any application in science or engineering would be welcome.