Reflectors should be your second nature
Reflectors are a class of matrices that are not introduced in all linear algebra textbooks. However, the book of Carl D. Meyer uses this class of matrices heavily for various fundamental results. Indeed, the book uses reflectors for theoretical reasons, such as proving the existence of fundamental transformations like SVD, QR, Triangulation, or Hessenberg decompositions (more to come below), as well as applications, such as the implementation of the QR algorithm via the Householder transformation or solution of large-scale linear systems…
Why should you have few distinct eigenvalues
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 $$p(x) = \prod\limits_{i=1}^ S(x-\lambda_i)^{a_i},$$…