After reading this article about the coordinate free definition of trace and determinant I am interested if these ideas can also be applied to other matrix invariants like for example the (coefficients of the) characteristic polynomial or the Jordan normal form. I am mainly looking for book recommendations.
I understand that my question is rather vague, if this question is best asked somewhere else it can be deleted.
Once you have any coordinate-free definition of the determinant you're done; you get all the other coefficients of the characteristic polynomial $\det(t - A)$ and these coefficients freely generate all polynomial matrix invariants. One way to prove this is to argue that any polynomial matrix invariant must be a symmetric polynomial in the eigenvalues, then appeal to the fundamental theorem of symmetric polynomials to argue that such polynomials are freely generated by the elementary symmetric polynomials in the eigenvalues, which are exactly the coefficients of the characteristic polynomial.
Alternatively, once you have any coordinate-free definition of the trace, the $k^{th}$ coefficient of the characteristic polynomial is $(-1)^k \text{tr}(\Lambda^k(A))$ where $\Lambda^k$ is the $k^{th}$ exterior power. This can be proven by showing that when $A$ is diagonalizable a basis of eigenvectors of $\Lambda^k(A)$ is given by wedge products of $k$ linearly independent eigenvectors from a basis of eigenvectors of $A$.
It is really a crime, incidentally, to discuss the coordinate-free definition of the trace without drawing any pictures. The pictures are called string diagrams and they make what's going on here very transparent. You can see them here and here. Here is a teaser, the string diagram proof of the cyclicity of the trace $\text{tr}(AB) = \text{tr}(BA)$: