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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 Sylvesterequation $A^TS = SA$ and $B$ is a real positive definite diagonal matrix whosediagonal elements are distinct each other, and show the S-Oja-Brockett equationhas the global convergence to eigenvalues and its eigenvectors of $A$.

Dynamical systems for eigenvalue problems of axisymmetric matrices with positive eigenvalues