1 / 17

Systems of Linear Equations (see Appendix A.6, Trucco & Verri)

Systems of Linear Equations (see Appendix A.6, Trucco & Verri). CS485/685 Computer Vision Prof. George Bebis. Systems of linear equations. An arbitrary system of m linear equations in n unknowns can be written as:. where:. or. Example (m=n=2):. Solutions of Ax=b (m = n).

obert
Download Presentation

Systems of Linear Equations (see Appendix A.6, Trucco & Verri)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Systems of Linear Equations(see Appendix A.6, Trucco & Verri) CS485/685 Computer Vision Prof. George Bebis

  2. Systems of linear equations • An arbitrary system of m linear equations in n unknowns can be written as: where: or Example (m=n=2):

  3. Solutions of Ax=b (m = n) • Characterize the solutions of Ax=b using conditions on the rank of A and A|b (i.e., augmented matrix).

  4. Solutions of Ax=b (m=n) • The system has one solution if : rank(A|b) = rank(A) = n Solution: i.e., b be expressed as a linear combination of the columns of A:

  5. Solutions of Ax=b (m=n) The following statements are equivalent: (a)rank(A|b) = rank(A) = n (b) A is invertible (c) (d) b has a unique expansion in the column space of A

  6. Solving Ax=b using SVD • Assuming that A=UDVT, then UDVTx=b or DVTx=UTb • Setting VTx=z and UTb=d, we have Dz=d (1) Compute z=D-1d (i.e., assume no zeroes in the diagonal) (2) Compute solution x =Vz Ax=b

  7. Solutions of Ax=b (m=n) (2) The system has no solutionif rank(A|b) > rank(A) b cannot be expressed as a linear combination of the columns of A e.g., using substitution leads to the contradiction 16=9

  8. Solutions of Ax=b (m=n) (3) The system has infinitely many solutionsif rank(A|b) = rank(A) < n - Less equations than unknowns (i.e, free variables). - b can be expressed as a linear combination of the columns of A in more than one ways.

  9. Homogeneous system: Ax=0 (m = n) • If b=0, then Ax=0 is called homogeneous. (1) Has the trivial solution x=0 iff rank(A) = n (i.e., A is invertible) (2) Has a non-trivial solution iff rank(A) < n (i.e., A is singular)

  10. m>n m<n Over/Under determined Systems

  11. Solving Ax=b (m > n) • Consider the over-determined system of linear equations: • Let r be the residual vector for some x: • The vector x* which yields the smallest possible residual is called a least-squaressolution:

  12. Solving Ax=b (m > n) (cont’d) • Although a least-squares solution always exist, it might not be unique! • The least-squares solution x with the smallest norm ||x|| is unique and it is given by:

  13. Solving Ax=b (m > n) - Example :

  14. Computing A+ using SVD • If ATA is ill-conditioned (or singular), we can use SVD to obtain a least squares solution as follows: where: (where t is a small threshold)

  15. Homogeneous systems • The minimum-norm solution is x=0; need to modify the meaning of a least-squares solution by imposing the constraint: • This is a "constrained" optimization problem:

  16. Homogeneous systems (cont’d) • The solution for homogeneous systems is not always unique. • Special case: Solution: (vnis the last column of V; the one corresponding to the smallest σ)

  17. Homogeneous systems (cont’d) • General case: Solution: with (vn-k+1, …,vnare the last columns of V ; correspond to the smallest σ’s)

More Related