Homework1_2011 - A is positive denite, then A 1 is...

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ESE 415 Optimization Assignment 1 Due: February 2, 2011 1. Consider the function f ( x ) defined on R 2 by f ( x, y ) = x 3 + e 3 y 3 xe y . Show that f ( x ) has exactly one critical point and that this point is a local minimizer but not a global minimizer of f ( x ). 2. Eigenvalues and eigenvectors play important roles in optimization. (a) Find eigenvalues and eigenvectors for the following matrices: (i) [ 0 2 2 2 ] , (ii) [ 20 5 5 1 ] , (iii) 3 2 2 2 7 2 2 2 3 (b) Let A be an n × n matrix and let λ 1 , . . . , λ n be the eigenvalues of A . λ ( A ) denotes the set of eigenvalues of A . Show the following propositions described in class. 1. λ ( cI + A ) = { c + λ 1 , . . . , c + λ n } , where I is the identity matrix. 2. λ ( A k ) = { λ k 1 , . . . , λ k n } . 3. If A is nonsingular, then λ ( A 1 ) = { 1 λ 1 , . . . , 1 λ n } . 4. λ ( A ) = λ ( A T ). 3. Let A S n be an n × n symmetric matrix. (a) Suppose that v i ’s are the normalized eigenvectors of A , i.e., v i = 1 for all i = 1 , . . . , n . Show that A = n i =1 λ i v i v T i , where λ i are the eigenvalues corresponding to v i . (b) Show that if
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Unformatted text preview: A is positive denite, then A 1 is symmetric and positive denite. (c) Classify the following matrices according to whether they are positive or negative denite or semidenite or indenite: ( a ) 1 0 0 0 3 0 0 0 5 . ( b ) 3 1 2 1 5 3 2 3 7 . ( c ) 2 4 4 8 3 . 1 4. Let A be a square n n matrix. (a) Show that A + A T is symmetric. (b) Show that x T Ax = x T ( A + A T 2 ) x for all x R n . Conclude that x T Ax 0 for all x R n if and only if the symmetric matrix A + A T is positive semidenite. 5. Dene f : R 2 R by setting f (0) = 0 and f ( x, y ) = xy x 2 + y 2 if ( x, y ) = 0 . For which vectors d = 0 does f (0; d ) exist? Evaluate it when it exists. 2...
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Homework1_2011 - A is positive denite, then A 1 is...

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