Dynamical systems for eigenvalue problems of axisymmetric matrices with positive eigenvalues

• URL: http://arxiv.org/abs/2307.09635v1
• Date: Thu, 29 Jun 2023 20:39:55 GMT
• Title: Dynamical systems for eigenvalue problems of axisymmetric matrices with positive eigenvalues
• Authors: Shintaro Yoshizawa
• Abstract summary: We show the S-Oja-Brockett equation has the global convergence to eigenvalues and its eigenvectors of $A$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the eigenvalues and eigenvectors of an axisymmetric matrix$A$ with some special structures. We propose S-Oja-Brockett equation $\frac{dX}{dt}=AXB-XBX^TSAX,$ where $X(t) \in {\mathbb R}^{n \times m}$ with $m \leq n$, $S$ is a positive definite symmetric solution of the Sylvester equation $A^TS = SA$ and $B$ is a real positive definite diagonal matrix whose diagonal elements are distinct each other, and show the S-Oja-Brockett equation has the global convergence to eigenvalues and its eigenvectors of $A$.

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