Content deleted Content added
Citation bot (talk | contribs) Alter: template type, first2. Add: jstor, doi, eprint, class. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by SevenSpheres | Category:Articles with missing Cite arXiv inputs | #UCB_Category 101/107 |
ShelfSkewed (talk | contribs) |
||
Line 346:
\text{with Jacobian:}\,\mathbf{E_x}=\mathbf{T_x}\,\text{at}\,\tilde\mathbf x=\mathbf0.
</math>
Since the Jacobian is injective (full rank: <math>m</math>), a local (not necessarily unique) [[left inverse function|left inverse]], say <math>r^*_\mathbf x</math> with Jacobian <math>\mathbf R^*_\mathbf x</math>, exists such that <math>r^*_\mathbf x(e_\mathbf x(\tilde x))=\tilde x</math> and <math>\mathbf R^*_\mathbf x\mathbf{T_x}=\mathbf I_m</math>. In practice we do not need the left inverse function itself, but we ''do'' need its Jacobian, for which the above equation does not give a unique solution. We can however enforce a unique solution for the Jacobian by choosing the left inverse as, <math>r_\mathbf x:\R^n\to\R^m</math>:
:<math>
r_\mathbf x(\mathbf z) = r^*_\mathbf x(\pi(\mathbf z))\,,\text{with Jacobian:}\,
| |||