Some useful linear algebra

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

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

Similarity transform then A and B have the same eigenvalues The eigenvector x of A corresponds to the eigenvector M-1x 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 si2 are eigenvalues of ATA Columns of U (u1 , u2 , u3 ) are eigenvectors of AAT Columns of V (v1 , v2 , v3 ) are eigenvectors of ATA

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

Least squares solution of homogeneous equation Ax=0

Enforce orthonormality constraints on an estimated rotation matrix R’

Newton iteration f( ) is nonlinear parameter measurement

Levenberg Marquardt iteration