linear algebra - Geometric interpretation of characteristic polynomial

A rather simple response is to differentiate the characteristic polynomial and use your interpretation of the determinant.

$$det(I-tf) = {t^n}det(frac{1}{t}I-f) = (-t)^ndet(f-frac{1}{t}I)= {(-t)^n}chi(f)(1/t)$$

So if we let $chi(f)(t) = Sigma_{i=0}^n a_it^i$, then ${(-t)^n}chi(f)(1/t) = (-1)^nSigma_{i=0}^n a_it^{n-i}$

But $I-tf$ is the path through the identity matrix, and $Det(A)$ measures volume distortion of the linear transformation $A$.

$$det(I-tf)^{(k)}(t=0) = (-1)^nk!a_{n-k}$$

and a change of variables ($tlongmapsto -t$) gives (and superscript $(k)$ indicates $k$-th derivative)

$$det(I+tf)^{(k)}(t=0) = (-1)^{n+k}k!a_{n-k}$$

So the coefficients of the characteristic polynomial are measuring the various derivatives of the volume distortion, as you perturb the identity transformation in the direction of $f$.

$$a_k = frac{det(I+tf)^{(n-k)}(t=0)}{(n-k)!}$$