  Generalized eigenvalue problems (GEPs) play an important role in the variety
of fields including engineering, machine learning and quantum chemistry.
Especially, many problems in these fields can be reduced to finding the minimum
or maximum eigenvalue of GEPs. One of the key problems to handle GEPs is that
the memory usage and computational complexity explode as the size of the system
of interest grows. This paper aims at extending sequential quantum optimizers
for GEPs. Sequential quantum optimizers are a family of algorithms that
iteratively solve the analytical optimization of single-qubit gates in a
coordinate descent manner. The contribution of this paper is as follows. First,
we formulate the GEP as the minimization/maximization problem of the fractional
form of the expectations of two Hermitians. We then showed that the fractional
objective function can be analytically minimized or maximized with respect to a
single-qubit gate by solving a GEP of a 4 $\times$ 4 matrix. Second, we show
that a system of linear equations (SLE) characterized by a positive-definite
Hermitian can be formulated as a GEP and thus be attacked using the proposed
method. Finally, we demonstrate two applications to important engineering
problems formulated with the finite element method. Through the demonstration,
we have the following bonus finding; a problem having a real-valued solution
can be solved more effectively using quantum gates generating a complex-valued
state vector, which demonstrates the effectiveness of the proposed method.
