Eigensystem Analysis

An ordinary linear eigensystem problem is represented by the equation Ax = λ x, where A denotes an n × n matrix. The value λ is an eigenvalue and x ≠ 0 is the corresponding eigenvector. The eigenvector is determined up to a scalar factor. In all eigenvalue functions, we have chosen this factor so that x has Euclidean length one, and the component of x of largest magnitude is positive. The eigenvalues and corresponding eigenvectors are sorted, then returned in the order of largest to smallest complex magnitude. If x is a complex vector, this component of largest magnitude is scaled to be real and positive. The entry where this component occurs can be arbitrary for eigenvectors having non-unique maximum magnitude values.