Understanding Linear Algebra: Vectors, Eigenvalues, and Solving Equations

1. Some useful linear algebra

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

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

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

6. Rank and Nullspace

7. 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

10. 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

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

13. Least squares solution of homogeneous equation Ax=0

14. Enforce orthonormality constraints on an estimated rotation matrix R’

15. Newton iteration f( ) is nonlinear parameter measurement

16. Levenberg Marquardt iteration