We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as the bidiagonal singular value decomposition and the symmetric eigenvalue decomposition. The advantages of this method stand for general normal matrices which include orthogonal matrices. We provide a stability and a complexity analysis of the method. The numerical performance is compared with existing algorithms. In most cases, the method has the same operation count as the Hessenberg factorization of a dense matrix. Finally, we provide experiments for the application of a Karcher mean on the special orthogonal group.

翻译：本文提出一种通过分解稠密实正规矩阵$A$的斜对称部分来计算其特征值分解的新方法。该方法依赖于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。本方法的优势适用于包含正交矩阵在内的广义正规矩阵。我们对该方法的稳定性和计算复杂度进行了分析，并与现有算法的数值性能进行了比较。在多数情况下，该方法具有与稠密矩阵海森伯格分解相同的运算量级。最后，我们给出了该方法在特殊正交群上Karcher均值计算中的应用实验。