I am a high schooler who has just recently learned a bit about matrices and the Cayley-Hamiltonian theorem. And my teachers have told me that to get the characteristic equation of many matrix $A$, I have to solve for det.$[A-\lambda I] = 0$ where $I$ is the identity matrix of same order as that of $A$.
So what does it mean geometrically for a matrix $A$ to satisfy the cayley hamiltonian theorem ? like if the above mentioned determinant is zero then as per my understanding, if we apply the above matrix as a linear transformation then it will squish everything down to just a single line.
What does it mean geometrically when we get a value of lambda that isn't purely real ? Say for example if we get $\lambda = 3 + 4\iota$, then what would that mean ?