2.6 Strong Interaction of Two Eigenvalues

Let us consider matrix family (2.62). Its eigenvalues plotted in the three-dimensional space (Re ?, Im ?, p) and the complex plane are shown in Fig. 2.2, where the arrows indicate motion of the eigenvalues with an increase of p. The interaction is described by two identical parabolae lying in perpendicular planes. With an increase of p the eigenvalues approach along the imaginary axis on the complex plane, collide, and then diverge along the real axis in different directions. Such interaction is typical for a double eigenvalue ? ₀ with a single eigenvector. We call it strong interaction.

Fig. 2.2: Strong interaction of eigenvalues for matrix family (2.62).

Let us consider an arbitrary matrix family A(p). Let ? ₀ be a double nonderogatory eigenvalue ? ₀ of the matrix A ₀ =A( p ₀) with corresponding right and left Jordan chains u ₀, u ₁ and v ₀, v ₁ satisfying equations (2.64) and normalization conditions (2.65). Our aim is to study behavior of two eigenvalues ?, which are coincident and equal to ? ₀ at p ₀, with a change of the vector of parameters p in the vicinity of the initial point p ₀. For this purpose we assume a variation p= p ₀ + ?e, where e=( e ₁, ..., e _n) is a vector of...