2.8 Bifurcation of Double Semi-Simple Eigenvalue

Let us consider the two-parameter matrix family

(2.156)

Eigenvalues of the matrix A(p) are

(2.157)

At p ₀=0 the matrix A ₀= A( p ₀) has the semi-simple double zero eigenvalue. We see that eigenvalues (2.157) are not differentiate functions of the parameters at p ₀. Nevertheless, leaving a single parameter, for example, setting p ₂=0, we get two differentiate functions for the eigenvalues

(2.158)

Thus, directional derivatives of the semi-simple eigenvalue exist.

Now let us consider an arbitrary family of matrices A(p). Let p ₀ be a point in the parameter space, where the matrix A ₀= A( p ₀) has a semi-simple double eigenvalue ? ₀. There are two linearly independent eigenvectors u ₁ and u ₂ satisfying the equations

(2.159)

Two left eigenvectors satisfy the equations

(2.160)

and can be uniquely determined for given u ₁, u ₂ by the normalization conditions

(2.161)

Assuming perturbation of the parameter vector along curve (2.67), we can express perturbations of the eigenvalue ? ₀ and corresponding eigenvectors in the form of power series [Vishik and Lyusternik (1960)]

(2.162)

Notice that any linear combination of the eigenvectors u ₁ and u ₂ is also an eigenvector. This means that the zero order term w ₀ in the expansion for the eigenvector u (the limit value of the eigenvector u as ? ?