We propose a fast method for computing the eigenvalue decomposition of a dense real normal matrix $A$. The method leverages algorithms that are known to be efficiently implemented, such as the bidiagonal singular value decomposition and the symmetric eigenvalue decomposition. For symmetric and skew-symmetric matrices, the method reduces to calling the latter, so that its advantages are for orthogonal matrices mostly and, potentially, any other normal matrix. The method relies on the real Schur decomposition of the skew-symmetric part of $A$. To obtain the eigenvalue decomposition of the normal matrix $A$, additional steps depending on the distribution of the eigenvalues are required. We provide a complexity analysis of the method and compare its numerical performance with existing algorithms. In most cases, the method is as fast as obtaining the Hessenberg factorization of a dense matrix. Finally, we evaluate the method's accuracy and provide experiments for the application of a Karcher mean on the special orthogonal group.

翻译：我们提出了一种计算稠密实正规矩阵$A$特征值分解的快速方法。该方法利用了已知可高效实现的算法，如双对角奇异值分解和对称特征值分解。对于对称矩阵和斜对称矩阵，该方法简化为调用后者，因此其主要优势体现在正交矩阵上，并可能适用于其他正规矩阵。该方法依赖于$A$斜对称部分的实舒尔分解。为获得正规矩阵$A$的特征值分解，需要根据特征值分布进行附加步骤。我们提供了该方法的复杂度分析，并将其数值性能与现有算法进行比较。在多数情况下，该方法的速度与获得稠密矩阵的海森伯格分解相当。最后，我们评估了方法的精度，并在特殊正交群上进行了Karcher均值应用的实验。