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hw4_2009_10_15_01

hw4_2009_10_15_01 - EE263 S Lall 2009.10.15.01 Homework 4...

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EE263 S. Lall 2009.10.15.01 Homework 4 Due Thursday 10/22. 1. A Pythagorean inequality for the matrix norm. 55035 Suppose that A R m × n and B R p × n . Show that vextenddouble vextenddouble vextenddouble vextenddouble bracketleftbigg A B bracketrightbiggvextenddouble vextenddouble vextenddouble vextenddouble radicalbig bardbl A bardbl 2 + bardbl B bardbl 2 . Under what conditions do we have equality? 2. Eigenvalues and singular values of a symmetric matrix. 55080 Let λ 1 , . . . , λ n be the eigenvalues, and let σ 1 , . . . , σ n be the singular values of a matrix A R n × n , which satisfies A = A T . (The singular values are based on the full SVD: If rank( A ) < n , then some of the singular values are zero.) You can assume the eigenvalues (and of course singular values) are sorted, i.e. , λ 1 ≥ · · · ≥ λ n and σ 1 ≥ · · · ≥ σ n . How are the eigenvalues and singular values related? 3. Degenerate ellipsoids 0176 The picture below shows a degenerate ellipsoid. -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 In two dimensions, a degenerate ellipsoid is a slab ; the sides are parallel lines. For the above example the slab has half-width 1 (i.e., it has width 2) and the center axis points in the direction v = bracketleftbigg 2 1 bracketrightbigg We’ll call the slab S . (a) Find a symmetric matrix Q R 2 × 2 such that the slab above is S = braceleftBig x R 2 | x T Qx 1 bracerightBig (b) Is Q positive definite? 1

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EE263 S. Lall 2009.10.15.01 (c) Consider the matrix P = bracketleftbigg 0 . 04 0 . 06 0 . 06 0 . 09 bracketrightbigg Plot the slab corresponding to P . (Sketch it by hand, if you prefer.) What is the axis of the slab? (i.e., the long axis). What is the half-width of the slab?
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hw4_2009_10_15_01 - EE263 S Lall 2009.10.15.01 Homework 4...

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