Linear Algebra and the C Language/a0nf

__NOTOC__ Theory: Eigenvalues ​​for programmers In Mathematics Let A be a matrix. Let T be a transformation that maps every vector v to the vector Av. T(v) = Av This transformation can perform rotations, translations, expansions, etc., on the vectors v. A particular transformation is one that only modifies the length of the vector and lets its direction unchanged (or reversed). T(v) = Av T(v) = λv In this particular case, we say that λ is an eigenvalue, and v is an eigenvector of this transformation. We have : T(v) = Av T(v) = λv Then : Av = λv 0 = λv - Av Transfer Av 0 = λ Id v - Av Introduce the identity matrix 0 = (λ Id - A) v Factor v 0 = B v Introduce B = (λ Id - A) 0 = B v If B is Invertible ....... PROBLEM invB 0 = invB B v Simplify 0 = id v 0 = v This stops the algorithm (v=0) 0 = B v If B is not invertible ... SOLUTION det(B) = 0 The determinant of B = 0 det(λ Id - A) = 0 That gives In mathematics, this allows us to obtain the characteristic polynomial. det(λ Id - A) = 0 det(A - λ Id) = 0 | λ-a -b | | a-λ b | | | = 0 | | = 0 | -c λ-d | | c d-λ | Expanding the determinant will give us a polynomial which will allow us to calculate the eigenvalues ​​of A. In programming We will use an iterative algorithm, which utilizes QR decomposition, to calculate the eigenvalues ​​of A.