A family of symmetric matrices $A_1,\ldots, A_d$ is SDC (simultaneous diagonalization by congruence) if there is an invertible matrix $X$ such that every $X^T A_k X$ is diagonal. In this work, a novel randomized SDC (RSDC) algorithm is proposed that reduces SDC to a generalized eigenvalue problem by considering two (random) linear combinations of the family. We establish exact recovery: RSDC achieves diagonalization with probability $1$ if the family is exactly SDC. Under a mild regularity assumption, robust recovery is also established: Given a family that is $\epsilon$-close to SDC then RSDC diagonalizes, with high probability, the family up to an error of norm $\mathcal{O}(\epsilon)$. Under a positive definiteness assumption, which often holds in applications, stronger results are established, including a bound on the condition number of the transformation matrix. For practical use, we suggest to combine RSDC with an optimization algorithm. The performance of the resulting method is verified for synthetic data, image separation and EEG analysis tasks. It turns out that our newly developed method outperforms existing optimization-based methods in terms of efficiency while achieving a comparable level of accuracy.

翻译：给定一族对称矩阵 $A_1,\ldots, A_d$，若存在可逆矩阵 $X$ 使得每个 $X^T A_k X$ 均为对角矩阵，则称该矩阵族可通过合同变换同时对角化（SDC）。本文提出一种新型随机SDC（RSDC）算法，通过考虑该族矩阵的两个（随机）线性组合，将SDC问题转化为广义特征值问题。我们建立了精确恢复性质：若矩阵族严格满足SDC条件，RSDC以概率 $1$ 实现对角化。在温和正则性假设下，还建立了鲁棒恢复性质：对于 $\epsilon$-接近SDC的矩阵族，RSDC以高概率将其对角化至范数误差为 $\mathcal{O}(\epsilon)$。在正定性假设（实际应用中常成立）下，得到了更强结果，包括变换矩阵条件数的界。面向实际应用，我们建议将RSDC与优化算法相结合。在合成数据、图像分离和脑电信号分析任务中验证了该方法的性能。结果表明，新方法在实现相当精度的同时，效率优于现有基于优化的方法。