Solution of the matrix eigenvalue problem with applications to the study of free linear dynamical systems

The new idea is to study the stability behavior of the solution x = x ( t ) of the initial value problem x ˙ = Ax , t ⩾ t 0 , x ( t 0 ) = x 0 , with A ∈ C n × n , in a weighted (semi-) norm ∥ · ∥ R where R is taken as an appropriate solution of the matrix eigenvalue problem RA + A * R = ρ R , rather than as the solution of the algebraic Lyapunov matrix equation RA + A * R = - S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues λ i ( A ) have negative real parts, i.e., Re λ i ( A ) < 0 , i = 1 , … , n , then ρ = ρ i = 2 Re λ i ( A ) < 0 , the associated eigenmatrices R = R i are positive semi-definite, and ∥ x ( t ) ∥ R i = ∥ x 0 ∥ R i e Re λ i ( A ) ( t - t 0 ) → 0 ( t → ∞ ) , which is much more than the old result, which only states that lim t → ∞ x ( t ) = 0 . Especially, the semi-norms ∥ · ∥ R i have a decoupling and filter effect on x ( t ) . Further, new two-sided bounds (depending on x 0 ) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings.