A Computable Spectral Theorem

Computing the spectral decomposition of a normal matrix is among the most frequent tasks to numerical mathematics. A vast range of methods are employed to do so, but all of them suffer from instabilities when applied to degenerate matrices, i.e., those having multiple eigenvalues. We investigate the spectral representation’s effectivity properties on the sound formal basis of computable analysis. It turns out that in general the eigenvectors cannot be computed from a given matrix. If however the size of the matrix’ spectrum (=number of different eigenvalues) is known in advance, it can be diagonalized effectively. Thus, in principle the spectral decomposition can be computed under remarkably weak non-degeneracy conditions.

• Type: Conference
• Year: 2000
• Where: CCA
• Authors: Martin Ziegler, Vasco Brattka
• Keywords: CCA 2000, Normal Matrix, Spectral Decomposition, Spectral Representation Effectivity, Theoretical Computer Science