The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or Jordan normal forms. Perturbation theory and various regularization techniques help address some numerical difficulties of computation eigenvectors, but often rely on per se uncomputable quantities, such as a minimal gap between eigenvalues. In this note, the eigenvalue problem is revisited within constructive analysis allowing to explicitly consider numerical uncertainty. Exact eigenvectors are substituted by approximate ones in a suitable format. Examples showing influence of computation precision are provided.

特征值问题在线性代数及其在控制和优化方法中的应用中起着核心作用。特别地，许多矩阵分解依赖于特征值-特征向量对的计算，例如对角线或约旦法线形式。摄动理论和各种正则化技术有助于解决计算特征向量的一些数值困难，但通常依赖于本身不可计算的量，例如特征值之间的最小间隙。在本说明中，特征值问题在构造分析中得到了重新考虑，从而可以明确考虑数值不确定性。确切的特征向量将以合适的格式替换为近似的特征向量。提供了显示计算精度影响的示例。