characteristic polynomial

The addition and subtraction of two polynomials \(P(x)\) and \(Q(x)\) are fundamental mathematical operations frequently occurring in various applications.

Eigenvalues and eigenvectors of a square matrix: definition via \(A\mathbf{v} = \lambda\mathbf{v}\), characteristic equation \(\det(A – \lambda I) = 0\), eigenspaces, and diagonalization.

A matrix is diagonalizable if there exists a basis of eigenvectors and it can be written in the form \( A = P D P^{-1} \) with \( D \) diagonal.