The minimum polynomial initially looks awfully similar to the characteristic polynomial, and it is not clear if learning it will have any practical utility. But it does. In fact, one of the most popular and efficient optimization techniques, namely conjugate gradients, relies on the Krylov sequences, which is build upon the concept of minimum polynomial. First, let’s see the how the characteristic and minimum polynomials are defined. The characteristic polynomial $p(x)$ of a matrix $\mathbf A$ is $$p(x) = \prod\limits_{i=1}^ S(x-\lambda_i)^{a_i},$$…