Some useful linear algebra

Linearly independent vectors span(V): span of vector space V is all linear combinations of vectors v i, i.e.

The eigenvalues of A are the roots of the characteristic equation Eigenvectors of A are columns of S diagonal form of matrix

Similarity transform then A and B have the same eigenvalues The eigenvector x of A corresponds to the eigenvector M -1 x of B

Rank and Nullspace

Least Squares More equations than unknowns Look for solution which minimizes ||Ax-b|| = (Ax-b) T (Ax-b) Solve Same as the solution to LS solution

Properties of SVD Columns of U (u 1, u 2, u 3 ) are eigenvectors of AA T Columns of V (v 1, v 2, v 3 ) are eigenvectors of A T A   2 are eigenvalues of A T A

with equal to for all nonzero singular values and zero otherwise pseudoinverse of A Solving

Least squares solution of homogeneous equation Ax=0

Enforce orthonormality constraints on an estimated rotation matrix R’

Newton iteration measurement parameter f( ) is nonlinear

Levenberg Marquardt iteration