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Lecture 5 Introduction to Linear Algebra Shankar Sastry September 13 th , 2005

Lecture 5 Introduction to Linear Algebra Shankar Sastry September 13 th , 2005. Orthogonal group. What is the set of transformations that preserve the inner product? Remember inner product under a transformation? More on this later …. Gram-Schmidt orthogonalization. MEMENTO!

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Lecture 5 Introduction to Linear Algebra Shankar Sastry September 13 th , 2005

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  1. Lecture 5 Introduction to Linear Algebra Shankar Sastry September 13th, 2005 Invitation to 3D vision

  2. Orthogonal group • What is the set of transformations that preserve the inner product? • Remember inner product under a transformation? • More on this later … Invitation to 3D vision

  3. Gram-Schmidt orthogonalization MEMENTO! will appear in calibration (aka Q-R) Structure of the Parameter matrix Invitation to 3D vision

  4. Structure induced by a linear map A X X’ Ra(A) T T Ra(A ) Nu(A) T T Nu(A ) Ra(A) Nu(A) Invitation to 3D vision

  5. Eigenvalues and eigenvectors • Eigenvalues and eigenvectors encode the “essence” of the linear map represented by A: the range space, the null space, the rank, the norm etc. • How do the notions of eigenvalues and eigenvectors generalize to NON-SQUARE matrices? • SVD, later … Invitation to 3D vision

  6. Symmetric matrices Invitation to 3D vision

  7. Symmetric matrices (contd.) Invitation to 3D vision

  8. Invitation to 3D vision

  9. Skew-symmetric matrices Invitation to 3D vision

  10. Skew-symmetric matrices (contd.) Invitation to 3D vision

  11. The singular value decomposition Invitation to 3D vision

  12. The SVD (contd.) Invitation to 3D vision

  13. The SVD: geometric interpretation A Invitation to 3D vision

  14. Pseudo-inverse and linear systems Invitation to 3D vision

  15. Fixed-rank approximation • Useful for matrix factorization • MEMENTO! Invitation to 3D vision

  16. Transformation groups Invitation to 3D vision

  17. Affine transformation • Not a linear transformation! • Can be made linear in HOMOGENEOUS COORDINATES MEMENTO! will appear everywhere Invitation to 3D vision

  18. Affine group (contd.) • Composition of affine transformations. • What is the inverse transformation? Invitation to 3D vision

  19. Orthogonal group • What is the set of transformations that preserve the inner product? • Remember inner product under a transformation? • More on this later … Invitation to 3D vision

  20. Euclidean group Invitation to 3D vision

  21. Unconstrained optimization Invitation to 3D vision

  22. Unconstrained optimization (contd.) Invitation to 3D vision

  23. Iterative minimization (local) • Steepest descent: • Newton’s method: • More in general: Invitation to 3D vision

  24. Gauss-Newton, Levemberg-Marquardt • Quadratic cost function • No second derivatives Invitation to 3D vision

  25. Constrained optimization Invitation to 3D vision

  26. Lagrangian function and multipliers Invitation to 3D vision

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