Homework1_2011

# Homework1_2011 - A is positive deﬁnite then A − 1 is...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A is positive deﬁnite, then A − 1 is symmetric and positive deﬁnite. (c) Classify the following matrices according to whether they are positive or negative deﬁnite or semideﬁnite or indeﬁnite: ( 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 semideﬁnite. 5. Deﬁne 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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Homework1_2011 - A is positive deﬁnite then A − 1 is...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online