Positive Definite $B$

We reduce it to an equivalent standard HEP. It is done as follows. Choose a decomposition for $B$:

$$B=GG^*.$$ (90)

Then the generalized eigenvalue problem for $A - \lambda B$ is equivalent to the standard HEP for $G^{-1}AG^{-*}$. Both share the same eigenvalues since

$$Ax=\lambda B x \quad\Leftrightarrow\quad G^{-1}AG^{-*} (G^* x)=\lambda (G^* x),$$

which also says that if $x$ is an eigenvector for the pair, $G^* x$ is an eigenvector for the matrix $G^{-1}AG^{-*}$, and on the other hand if $y$ is an eigenvector for $G^{-1}AG^{-*}$, $G^{-*}y$ is an eigenvector for the pair. Common choices for $G$ are:

1. $G=B^{1/2}$, the unique positive definite square root of $B$. In this case, $G^*=G$. This choice is good enough for theoretical investigations.
2. $G$ is the Cholesky factor; optionally with pivoting, i.e., $G$ is lower triangular with positive diagonal entries. This choice is preferred for numerical computations.
3. Analogously $G$ is upper triangular with positive diagonal entries. It shares the same advantage of the second choice.

In what follows, sometimes it is more convenient to use the inner product $(\cdot\,,\,\cdot)_M$ induced by a positive definite matrix $M$, the corresponding vector norm $\Vert\cdot\Vert _M$, and the two-vector angle function (more precisely, angle between the subspaces spanned by two vectors) $\theta_M(\,\cdot\,,\,\cdot\,)$. In our case, $M=B$ or $B^{-1}$. They are defined as follows.

$$\begin{eqnarray*} (x,y)_M & \equiv & y^*Mx, \\ \Vert x\Vert _M & \equiv & \sqrt... ...iv & \frac {\vert(x,y)_M\vert}{\Vert x\Vert _M\Vert y\Vert _M}. \end{eqnarray*}$$

When $M=I$, all three reduce to the usual definitions. It is rather easy to see that

$$\Vert M^{-1}\Vert _2^{-1/2} \Vert x\Vert _2\le\Vert x\Vert _M\le\Vert M\Vert _2^{1/2} \Vert x\Vert _2.$$ (91)

With some extra work, we can relate $\theta_M$ to the usual angle function, e.g., for $M=I$, as follows.

$$(2\kappa(M))^{-1/2} \sin\theta(x,y) \le\sin\theta_M(x,y) \le (2\kappa(M))^{1/2}\sin\theta(x,y).$$ (92)