The process of finding an unknown scalar, λ and a nonzero vector, x for a given non zero square matrix, A of dimension n x n is called matrix eigenvalue or eigenvalue. Ax = λ x The λ and x which satisfies the above equation is called eigen value and eigenvector. Ax should be proportional to x. The multiplication will produce a new vector that will have the same or opposite direction as the original vector. The set of all the eigenvalues of A is called the spectrum of A. The largest of the absolute values of the eigenvalues of A is called the spectral radius of A. To determine eigenvalue and eigenvector, the equation can be written in matrix notation, (A - λI)x = 0 By Cramer's theorem, the homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero. A - λI is called characteristic matrix and D( λ) is characteristic determinant of A. The above equation is called characteristic ...