masks © 2004 invitation to 3d vision lecture 5 introduction to linear algebra shankar sastry...
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MASKS © 2004 Invitation to 3D vision
Lecture 5Lecture 5
Introduction to Linear AlgebraIntroduction to Linear AlgebraShankar SastryShankar Sastry
September 13September 13thth, 2005, 2005
MASKS © 2004 Invitation to 3D vision
What is the set of transformations that preserve the inner product?
Remember inner product under a transformation?
Orthogonal group
MASKS © 2004 Invitation to 3D vision
MEMENTO!will appear incalibration(aka Q-R)Structure of theParameter 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
COORDINATESMEMENTO!will appeareverywhere
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?
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