An image-processing routine applies a plane shear to the (dimensionless) pixel…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

An image-processing routine applies a plane shear y=Ax\mathbf{y} = \mathsf{A}\mathbf{x} to the (dimensionless) pixel coordinates of a photograph, with A=(1101).\mathsf{A} = \begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}. ((a)) Find the singular values of A\mathsf{A} from the eigenvalues of ATA\mathsf{A}^{\rm T}\mathsf{A}. ((b)) State the rank of A\mathsf{A}, its 2-norm ∥A∥2\|\mathsf{A}\|_2, and its condition number κ2=σ1/σ2\kappa_2 = \sigma_1/\sigma_2.

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