Eigen Value and Vecotr⚓︎
Definition : Consider a square matrix \(A=[a_{ij}]_{n{\times}n}\) and a nonzero vector \(v\) of length \(n\) Then there exit a scalar \(\lambda \in R\) such that \(Av = \lambda v\) . Where \(v\) is called eigenvector corresponding to eigenvalue \(\lambda\) of matrix \(A\).
Remark : 1. Linear transformation \(T : R^n \to R^n\) is equivalent to the square matrix \(A\) of order \(n \times n\). thus given a basis set of the vector space can be defined as set of eigen vectors of matrix \(A\) for linear transformation \(T\).
- Eigenvectors and eigenvalues exits in a wide range of applications like stability and vibration analysis of dynamical systems, atomic orbits, facial recognition and matrix diagonalization.
Faddeev-LeVerrier Algorithm⚓︎
Faddeev-LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial \(p_A(\lambda)=\det (\lambda I_n - A)\) of a square matrix, A. solving characteristic polynomial gives eigen values of matrix A as a roots of it and matrix polynomial of matrix A vanishes i.e p(A) = 0 by Cayley-Hamilton Theorem. Faddeev-Le Verrier algorithm works directly with coefficients of matrix \(A\).
Problem : Given a