Rayleigh–Ritz method: Difference between revisions

Revision as of 15:45, 23 May 2022

Revision as of 15:47, 23 May 2022

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=== Using the normal matrix ===

The definition of the singular value <math>\sigma</math> and the corresponding left and right singular vectors is <math>M v=\sigma u</math> and <math>M^* u=\sigma v</math>. Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the [[Hermitian matrix|Hermitian]] '''normal matrix''' <math> M^* M \in \mathbb{C}^{N \times N}</math> or <math> M M^* \in \mathbb{C}^{M \times M}</math>, whichever one is smaller size, one could determine the other set of left of right singular vectors simply by dividing by the singular values, i.e., <math>u = M v Mv/ \sigma</math> and <math>v = M^* u / \sigma</math>. However, the division is unstable or fails for small or zero singular values.

An alternative approach, e.g., defining the normal matrix as <math> A = M^* M \in \mathbb{C}^{N \times N}</math> of size <math>N</math>-by-<math>N</math>, takes advantage of the fact that for a given <math>N</math>-by-<math>m</math> matrix <math> W \in \mathbb{C}^{N \times m} </math> with [[orthonormal]] columns the eigenvalue problem of the Rayleigh–Ritz method for the <math>m</math>-by-<math>m</math> matrix