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MIT6_241JS11_lec05

# MIT6_241JS11_lec05 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 5: Matrix Perturbations Readings: DDV, Chapter 5 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 16, 2011 E. Frazzoli (MIT) Lecture 5: Matrix Perturbations Feb 16, 2011 1 / 10

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Outline 1 Matrix Perturbations E. Frazzoli (MIT) Lecture 5: Matrix Perturbations Feb 16, 2011 2 / 10
Introduction Important issues in engineering, and in systems and control science in particular, concern the sensitivity of computations, solution algorithms, design methods, to uncertainty in the input parameters. For example: What is the smallest perturbation (e.g., in terms of 2-norm) that makes a matrix singular? What is the impact on the solution of a least-square problem of uncertainty in the data? etc. E. Frazzoli (MIT) Lecture 5: Matrix Perturbations Feb 16, 2011 3 / 10

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�� Additive Perturbation Theorem (Additive Perturbation) Let A C m × n be a matrix with full column rank ( = n). Then min Δ C m × n {� Δ 2 : A + Δ has rank < n } = σ min ( A ) . Proof: If A + Δ has rank < n , then there exists x , with x 2 = 1, such that ( A + Δ) x = 0, i.e., Δ x = Ax . In terms of norms, Δ 2 ≥ � Δ x 2 = Ax 2 σ min ( A ) To prove that the bound is tight, let us construct a Δ that achieves it.
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MIT6_241JS11_lec05 - 6.241 Dynamic Systems and Control...

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