MASKS © 2004 Invitation to 3D vision Lecture 5 Introduction to Linear Algebra Shankar Sastry September 13 th, 2005

MASKS © 2004 Invitation to 3D vision What is the set of transformations that preserve the inner product? Remember inner product under a transformation? More on this later … Orthogonal group

MASKS © 2004 Invitation to 3D vision MEMENTO! will appear in calibration (aka Q-R) Structure of the Parameter matrix Gram-Schmidt orthogonalization

MASKS © 2004 Invitation to 3D vision Nu(A ) A T T Ra(A) Nu(A) Ra(A ) X X’ Nu(A) T T Ra(A) Structure induced by a linear map

MASKS © 2004 Invitation to 3D vision 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 … Eigenvalues and eigenvectors

MASKS © 2004 Invitation to 3D vision Symmetric matrices

MASKS © 2004 Invitation to 3D vision Symmetric matrices (contd.)

MASKS © 2004 Invitation to 3D vision

MASKS © 2004 Invitation to 3D vision Skew-symmetric matrices

MASKS © 2004 Invitation to 3D vision Skew-symmetric matrices (contd.)

MASKS © 2004 Invitation to 3D vision The singular value decomposition

MASKS © 2004 Invitation to 3D vision The SVD (contd.)

MASKS © 2004 Invitation to 3D vision A The SVD: geometric interpretation

MASKS © 2004 Invitation to 3D vision Pseudo-inverse and linear systems

MASKS © 2004 Invitation to 3D vision Useful for matrix factorization MEMENTO! Fixed-rank approximation

MASKS © 2004 Invitation to 3D vision Transformation groups

MASKS © 2004 Invitation to 3D vision Not a linear transformation! Can be made linear in HOMOGENEOUS COORDINATES MEMENTO! will appear everywhere Affine transformation

MASKS © 2004 Invitation to 3D vision Composition of affine transformations. What is the inverse transformation? Affine group (contd.)

MASKS © 2004 Invitation to 3D vision What is the set of transformations that preserve the inner product? Remember inner product under a transformation? More on this later … Orthogonal group

MASKS © 2004 Invitation to 3D vision Euclidean group

MASKS © 2004 Invitation to 3D vision Unconstrained optimization

MASKS © 2004 Invitation to 3D vision Unconstrained optimization (contd.)

MASKS © 2004 Invitation to 3D vision Iterative minimization (local) Steepest descent: Newton’s method: More in general:

MASKS © 2004 Invitation to 3D vision Gauss-Newton, Levemberg-Marquardt Quadratic cost function No second derivatives

MASKS © 2004 Invitation to 3D vision Constrained optimization

MASKS © 2004 Invitation to 3D vision Lagrangian function and multipliers