The continuous-time Rayleigh quotient flow on the sphere
Consider the eigenvalue problem for a symmetric matrix. The Rayleigh quotient can be defined by \(r_{A}(x)=x^{T}Ax\) where \(x\) is on the sphere \( x^{T}x=1,\) \ \(x\in \mathbb{R}^{n}.\) Using the projection onto the tangent space, the continuous time Rayleigh quotient flow is generated by \(\dot{x}=\pi _{x}\left( A-r_{A}(x)I_{n}\right) ^{-1}x\) where \(x(0)\in S^{n-1}.\) Thus the vector field is defined except when \(r_{A}(x)\) is an eigenvalue. The phase portrait is analyzed and it is shown that the flow of a nondegenerate solution visits each of the eigenvectors in an ascending order.
scientific article; zbMATH DE number 1956810
scientific article