Generalized Eigenvalue Problem

The generalized eigenvalue problem is a mathematical problem that arises in many areas of science and engineering. It is a generalization of the standard eigenvalue problem, where we seek the eigenvalues and eigenvectors of a square matrix. In the generalized eigenvalue problem, we consider two square matrices \(\symbf{A}\) and \(\symbf{B}\), and we seek the eigenvalues \(\lambda\) and eigenvectors \(\symbf{x}\) that satisfy the equation

\[\begin{equation}\label{eq:gen_eig} \symbf{A}\symbf{C}=\symbf{B}\symbf{C}\symbf{\Lambda}, \end{equation}\]

where \(\symbf{C}\) is a matrix of eigenvectors and \(\symbf{\Lambda}\) is a diagonal matrix of eigenvalues. The quick way to solve the generalized eigenvalue problem is to transform it into a standard eigenvalue problem by multiplying both sides of the equation by the inverse of \(\symbf{B}\) as

\[\begin{equation} \symbf{B}^{-1}\symbf{A}\symbf{C}=\symbf{C}\symbf{\Lambda}. \end{equation}\]

This method is not always numerically stable, especially when the matrices \(\symbf{A}\) and \(\symbf{B}\) are ill-conditioned. Should you try to use this method for the Roothaan equations in the Hartree–Fock method, you would find that the solution is is not correct. A more stable approach is to modify the equation \eqref{eq:gen_eig} as

\[\begin{align} \symbf{A}\symbf{C}&=\symbf{B}\symbf{C}\symbf{\Lambda}\nonumber \\ \symbf{B}^{-\frac{1}{2}}\symbf{A}\symbf{C}&=\symbf{B}^{\frac{1}{2}}\symbf{C}\symbf{\Lambda}\nonumber \\ \symbf{B}^{-\frac{1}{2}}\symbf{A}\symbf{B}^{-\frac{1}{2}}\symbf{B}^{\frac{1}{2}}\symbf{C}&=\symbf{B}^{\frac{1}{2}}\symbf{C}\symbf{\Lambda}, \end{align}\]

where we solve the standard eigenvalue problem for the matrix \(\symbf{B}^{-\frac{1}{2}}\symbf{A}\symbf{B}^{-\frac{1}{2}}\) and obtain the eigenvectors \(\symbf{B}^{\frac{1}{2}}\symbf{C}\) and eigenvalues \(\symbf{\Lambda}\). The eigenvectors of the original problem are then given by \(\symbf{C}=\symbf{B}^{-\frac{1}{2}}\symbf{B}^{\frac{1}{2}}\symbf{C}\) and the eigenvalues are the same.