section6 - A maps vectors in R 3 . Hint: Consider the...

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EE221A Section 6 9/30/11 1 Singular Value Decomposition Exercise 1. Show that the eigenvalues of a Hermitian matrix are all real. Exercise 2. Show that AA * , for A C m × n , is positive semidefinite. Exercise 3. Consider a real unitary matrix U R 3 × 3 . Give a geometric interpretation of how U maps vectors x 7→ Ux . Exercise 4. Now, consider a general matrix A R 3 × 3 . Using the singular value decomposition, give a geometric interpretation of how
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Unformatted text preview: A maps vectors in R 3 . Hint: Consider the action of A on the unit sphere. 2 Existence and uniqueness Exercise 5. Do the following functions satisfy the Lipschitz condition? a) f ( x ) not continous in x b) f ( x ) = x 2 on [-3 , 7] c) f ( x ) = x 2 d) f ( x ) = x on [0 , 3] e) f ( x ) = x 2 + 5 f) f ( x ) = sin x 1...
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