Pertubation theory for 3x3 matrix

[Ton]

Some one knows a study material to diagonalize a matrix mass for 3 neutral scalar using perturbation theory like \begin{equation}
M^2=\left(\begin{array}{ccc}
2 \lambda_{\phi} v_{\phi}^2 &\lambda_{\phi \sigma } v_{\phi}v_{\sigma} & \lambda_{\phi\eta} v_{\phi} v_{\eta} \\
\lambda_{\phi \sigma} v_{\phi} v_{\sigma} & 2\lambda_{\sigma} v_{\sigma}^2 & \lambda_{\sigma \eta} v_{\sigma}v_{\eta} \\
\lambda_{ \phi\eta} v_{\phi} v_{\eta} & \lambda_{\sigma \eta} v_{\sigma} v_{\eta} & 2 \lambda_{\eta} v_{\eta}^2

\end{array}\right) .
\end{equation}
with $$v_{\phi}\ll v_{\sigma} \ll v_{\eta}$$ to obtain all, mix, eigenvectors. Thanks!

 

[LightHero]

One reference to a study material that provides a method of diagonalizing a matrix using perturbation theory is the book "Linear Algebra Problem Book" by Paul R. Halmos. This book provides step-by-step instructions on how to use perturbation theory to solve for all eigenvectors of a matrix. Additionally, the book includes sample problems and solutions to help readers understand and apply the theory to their own problems.