On the Global Convergence of the Toda Lattice for Real Normal Matrices and Its Applications to the Eigenvalue Problem

Source abstract

The asymptotic behavior of the Toda lattice, when acting on real normal matrices, is studied. It is shown that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigenvalue as its element and for complex-conjugate pairs of eigenvalues the associated block is of size $2 \times 2$ with the real part as its diagonal elements and the (negative) imaginary part as its off-diagonal elements. This result generalizes the well-known asymptotic behavior of Jacobi matrices and is consistent with that from the $QR$-algorithm.