Isospectral flows on a class of finite-dimensional Jacobi matrices

Abstract

We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes n x n zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e. features a right-hand side with a nested commutator of matrices and structurally resembles the double-bracket o.d.e. studied by R.W. Brockett in 1991. We prove that its solutions converge asymptotically, that the limit is block-diagonal, and above all, that the limit matrix is defined uniquely as follows: for n even, a block-diagonal matrix containing 2 x 2 blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these 2 x 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition. For odd, there is one additional 1 x 1 block containing a zero that is the top left entry of the limit matrix. The results presented here extend some early work of Kac and van Moerbeke.