  In this paper, we propose a gradient-based block coordinate descent (BCD-G)
framework to solve the joint approximate diagonalization of matrices defined on
the product of the complex Stiefel manifold and the special linear group.
Instead of the cyclic fashion, we choose a block optimization based on the
Riemannian gradient. To update the first block variable in the complex Stiefel
manifold, we use the well-known line search descent method. To update the
second block variable in the special linear group, based on four kinds of
different elementary transformations, we construct three classes: GLU, GQU and
GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We
establish the global and weak convergence of these three algorithms using the
\L{}ojasiewicz gradient inequality under the assumption that the iterates are
bounded. We also propose a gradient-based Jacobi-type framework to solve the
joint approximate diagonalization of matrices defined on the special linear
group. As in the BCD-G case, using the GLU and GQU classes of elementary
transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms and
establish their global and weak convergence. All the algorithms and convergence
results described in this paper also apply to the real case.
