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.

翻译：本文提出了一种基于梯度的块坐标下降（BCD-G）框架，用于解决定义在复Stiefel流形与特殊线性群乘积空间上的矩阵联合近似对角化问题。不同于循环方式，我们选择基于黎曼梯度的块优化策略。为更新复Stiefel流形上的第一个块变量，采用经典的线搜索下降法；为更新特殊线性群中的第二个块变量，基于四种不同类型的初等变换构造了三类变换族：GLU、GQU和GU，进而得到三种BCD-G算法：BCD-GLU、BCD-GQU和BCD-GU。在迭代序列有界的假设下，利用Łojasiewicz梯度不等式建立了这三种算法的全局收敛性与弱收敛性。同时提出基于梯度的Jacobi型框架，用于求解定义在特殊线性群上的矩阵联合近似对角化问题。类似于BCD-G情形，利用GLU和GQU类初等变换，重点研究了Jacobi-GLU和Jacobi-GQU算法，并建立了它们的全局收敛性与弱收敛性。本文所描述的所有算法及收敛性结果同样适用于实矩阵情形。