A STUDY OF ISOSPECTRAL FLOW ON BANDED MATRICES
Keywords:
isospectral flow, differential equations on manifolds, limit set, symmetric matrices, sparse matrices, banded matrices.DOI:
https://doi.org/10.17654/0972111822007Abstract
An isospectral flow manipulates a matrix preserving its spectrum. In this paper, we study a flow, originally introduced by Arsie and Ebenbauer [2], of the form $\dot{P}=\left[\left[P^{\prime}, P\right]_{d u}, P\right]$, where $\dot{P}$ is the time derivative of matrix $P, P^{\prime}$ is transpose of matrix $P$, and $\left[P^{\prime}, P\right]_{d u}$ is same as matrix $\left[P^{\prime}, P\right]$ on the upper triangular elements and all elements below the diagonal are zero. We extend results in [2] to the case of real banded (band) matrices. If the initial condition, $P(0)=$ $P_0$, is a banded matrix having lower bandwidth $p>0$ with a simple and real spectrum, then $P(t)$ converges as $t \rightarrow \infty$ to a banded symmetric matrix having bandwidth $p$, isospectral to $P_0$. Also, the limit point has the same sign pattern in the $p$ th subdiagonal elements as in $P_0$.
Received: August 10, 2022
Accepted: September 27, 2022
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