2.4 Perturbation of Simple Eigenvalue

Let us assume that the matrix A smoothly depends on a vector of real parameters p=( p ₁, ..., p _n). The function A(p) is called a multi-parameter family of matrices. Eigenvalues of the matrix family are continuous functions of the parameter vector. In this section we study behavior of a simple eigenvalue of the matrix family A(p).

Let ? (p) be a simple eigenvalue of the matrix A(p). Since ? is a simple root of characteristic equation (2.2), we have

(2.40)

Using inequality (2.40) and the implicit function theorem applied to characteristic equation (2.2), we find that the simple eigenvalue ? (p) of the matrix family A(p) smoothly depends on the parameter vector, and its derivatives with respect to parameters are equal to

(2.41)

The eigenvector u(p) corresponding to ? (p) is determined up to a nonzero scaling factor. This eigenvector determines a one-dimensional null-subspace of the matrix operator A(p) ? ? (p)I smoothly dependent on p. Hence, the eigenvector u(p) can be chosen as a smooth function of the parameter vector.

Let us consider a point p ₀ in the parameter space and assume that ? ₀ is a simple eigenvalue of the matrix A ₀ =A( p ₀). Taking the derivative with respect to p _i of both sides of eigenvalue problem (2.1), we find

(2.42)

where u ₀