# Isospectral flows on a class of finite-dimensional Jacobi matrices

Author(s):T. Sutter, D. Chatterjee, F. Ramponi, J. Lygeros |
Conference/Journal:Systems and Control Letters, vol. 62, no. 5, pp. 388-394, Also available at: http://arxiv.org/abs/1202.1618 |

Abstract:We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes $n\times n$ 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 $2\times 2$ blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these $2\times 2$ blocks have the same sign as the respective entries in the matrix employed as initial condition. For $n$ odd, there is one additional $1\times 1$ 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. Further Information |
Year:2013 |

Type of Publication:(01)Article | |

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@article{Sutter2013388, title = {Isospectral flows on a class of finite-dimensional Jacobi matrices}, author = {Tobias Sutter and Debasish Chatterjee and Federico A. Ramponi and John Lygeros}, journal = {Systems and Control Letters}, volume = {62}, number = {5}, pages = {388 - 394}, year = {2013}, issn = {0167-6911}, doi = {10.1016/j.sysconle.2013.02.004}, url = {http://www.sciencedirect.com/science/article/pii/S0167691113000339}, keyword = {Isospectral flow} } | |

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