Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^\text{T}$ for all $i$, where $\Sigma_i$'s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis (ISA). This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$'s. In this paper, we propose a ``bi-block diagonalization'' method to solve BJBDP, and establish sufficient conditions under which the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors' knowledge, existing numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.

翻译：考虑到一个固定的 $mathcal{C ⁇ C_i ⁇ i=1m美元平方基体,矩阵盲联组合对角化问题(BJBDP)将找到一个完整的一列排名矩阵($A),以便找到一个完整的一列排名矩阵($A),以便找到一个完整的一列矩阵($C_i=A\Sigma_iA ⁇ text{T}$T$$美元),在美元的所有一美元中,$Sigma_i$($Slum_i$)都是有尽可能多的对角区块的对角矩阵。BJBDP在独立的子空间分析(ISA)中发挥着重要作用。本文考虑了BJBDP的识别问题,即在什么条件下和用什么手段,我们可以确定一个完整的对二元的对角模型($Sigma_i$)和块的对角结构($Sigma_i$),特别是当美元有噪音的时候。在本文中,我们建议一个“双块对二块对二角化方法的解算法”方法,并且为完成这项任务创造了足够的条件。NJ模拟模拟模拟模拟模拟模拟验证了我们的理论结果。对于理论的解算法是没有理论方法。