Commuting Hermitian matrices may be simultaneously diagonalized by a common unitary matrix. However, the numerical aspects are delicate. We revisit a previously rejected numerical approach in a new algorithm called ‘do-one-then-do-the-other’. One of two input matrices is diagonalized by a unitary similarity, and then the computed eigenvectors are applied to the other input matrix. Additional passes are applied as necessary to resolve invariant subspaces associated with repeated eigenvalues and eigenvalue clusters. The algorithm is derived by first developing a spectral divide-and-conquer method and then allowing the method to break the spectrum into, not just two invariant subspaces, but as many as safely possible. Most computational work is delegated to a black-box eigenvalue solver, which can be tailored to specific computer architectures. The overall running time is a small multiple of a single eigenvalue-eigenvector computation, even on difficult problems with tightly clustered eigenvalues. The article concludes with applications to a structured eigenvalue problem and a highly sensitive eigenvector computation.

中文翻译：

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几乎对易的 Hermitian 矩阵的同时对角化：做一个然后做另一个

通勤厄米特矩阵可以同时被一个共同的酉矩阵对角化。但是，数字方面很微妙。我们在一种名为“做一个，然后做另一个”的新算法中重新审视了以前被拒绝的数值方法。两个输入矩阵之一通过酉相似性对角化，然后将计算出的特征向量应用于另一个输入矩阵。根据需要应用额外的通道来解析与重复特征值和特征值簇相关联的不变子空间。该算法是通过首先开发一种光谱分而治之的方法，然后允许该方法将光谱分成两个不变的子空间，而不只是两个不变的子空间，而是尽可能多的安全子空间来推导出来的。大多数计算工作委托给黑盒特征值求解器，它可以针对特定的计算机体系结构进行定制。总运行时间是单个特征值-特征向量计算的小倍数，即使是在具有紧密聚集的特征值的难题上也是如此。本文最后介绍了在结构化特征值问题和高度敏感的特征向量计算中的应用。