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Isospectral flows on a class of finite-dimensional Jacobi matrices


T. Sutter, D. Chatterjee, F. Ramponi, J. Lygeros

Systems and Control Letters, vol. 62, no. 5, pp. 388-394, Also available at:

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.

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	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 = {},
	keyword = {Isospectral flow}
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