hw4_2009_10_15_01 - EE263 S. Lall 2009.10.15.01 Homework 4...

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Unformatted text preview: 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 1 , . . ., n be the eigenvalues, and let 1 , . . ., 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. , 1 n and 1 n . How are the eigenvalues and singular values related? 3. Degenerate ellipsoids 0176 The picture below shows a degenerate ellipsoid.-8-6-4-2 2 4 6 8-6-4-2 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 Well 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?...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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hw4_2009_10_15_01 - EE263 S. Lall 2009.10.15.01 Homework 4...

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