MIT6_241JS11_lec05

MIT6_241JS11_lec05 - 6.241 Dynamic Systems and Control...

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Unformatted text preview: 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 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 Additive Perturbation Theorem (Additive Perturbation) Let A C m n be a matrix with full column rank ( = n). Then min C m n { 2 : A + has rank < n } = min ( A ) . Proof: If A + has rank < n , then there exists x , with x 2 = 1, such that ( A + ) x = 0, i.e., x = Ax ....
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MIT6_241JS11_lec05 - 6.241 Dynamic Systems and Control...

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