SINGULAR VALUE DECOMPOSITION: SHORT REVIEW

Any MxN-matrix $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ can be decomposed in the following way (Golub and Van Loan, 1989):

$$\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}\ =\ \dis... ...$\sim$}}{\displaystyle \mathop{\mbox{\bf V}}_{\mbox{$\sim$}}}^T$$ (1)

where $$\mathop{\mbox{\bf U}}_{\mbox{$\sim$}}$$ is an MxM orthogonal matrix of eigenvectors that span the data space, $$\mathop{\mbox{\bf V}}_{\mbox{$\sim$}}$$ is an NxN orthogonal matrix of eigenvectors that span the model parameters space and $$\mathop{\mbox{\bf L}}_{\mbox{$\sim$}}$$ is an MxN diagonal matrix whose diagonal elements are the singular values of $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$. The columns of $$\mathop{\mbox{\bf U}}_{\mbox{$\sim$}}$$ are the eigenvectors of $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}{\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}^T$$ and the columns of $$\mathop{\mbox{\bf V}}_{\mbox{$\sim$}}$$ are the eigenvectors of ${\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}^T {\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}$.

When a singular value is zero, the corresponding eigenvector in data (or model) space cannot be mapped into model (or data) space. When a singular value is not zero but is small compared with the largest one (large condition number), the contribution of the corresponding eigenvectors into the solution must be eliminated or attenuated (regularization) because the problem may become unstable.