{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

math-2011-note10

math-2011-note10 - Example 6 Suppose that we have a...

This preview shows pages 1–4. Sign up to view the full content.

Example 6. Suppose that we have a production function Q = kx a y b . Then, Q x = akx a - 1 y b , Q y = bkx a y b - 1 2 Q x y = abkx a - 1 y b - 1 , 2 Q y x = abkx a - 1 y b - 1 14 Some Linear Algebra 14.1 Definiteness of Quadratic Forms ( § 16.2) Definition 27. Let A be an m × m symmetric matrix, then A is (a) positive definite if x T A x > 0 for all x negationslash = 0 in R m , (b) positive semidefinite if x T A x 0 for all x in R m , (c) negative definite if x T A x < 0 for all x negationslash = 0 in R m , (d) negative semidefinite if x T A x 0 for all x in R m , (e) indefinite if x T A x > 0 for some x in R m and < 0 for some other x in R m . Q ( x , y ) = - 3 x 2 - 6 y 2 0 and strictly negative except at 0 . Hence, it is negative definite, and ( x , y ) = 0 is its global maximizer. A = ? Q 3 ( x , y ) = 3 x 2 - 6 y 2 is indefinite. A = ? Q 4 ( x , y ) = x 2 + 2 xy + y 2 is positive semidefinite. A = ? Q 5 ( x , y ) = - ( x - y ) 2 is negative semidefinite. A = ? 14.2 Symmetric Matrices ( § 23.7) Definition 28 (p.579, 582) . Let A be a m × m matrix, v negationslash = 0 be a m-vector and r be a number. If A v = r v , then v is called an eigenvector (or characteristic vector ) of A, and r is called an eigenvalue (or characteristic value ) of A. Example 7. Consider A = parenleftbigg 2 4 4 2 parenrightbigg , v = parenleftbigg 1 - 1 parenrightbigg , w = parenleftbigg 1 1 parenrightbigg . Then, A v = parenleftbigg - 2 2 parenrightbigg = - 2 v , and A w = parenleftbigg 6 6 parenrightbigg = 6 w . Hence, the eigenvalues of A are -2 and 6. Theorem 44 (Thm 23.16, p.621) . Let A be a symmetric m × m matrix. Then there is a nonsingular matrix P whose columns w 1 , ··· , w m are eigenvectors of A such that (i) w 1 , ··· , w m are mutually orthogonal to each other 40

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

View Full Document
(ii) P - 1 = P T , and (iii) P - 1 AP = P T AP = r 1 0 ··· 0 0 r 2 ··· 0 . . . . . . . . . . . . 0 0 ··· r m If we write the diagonal matrix as D , then we have A = PDP T . Hence, x T A x = x T PDP T x = y T D y , where y = P T x . A is positive(negative) definite D is positive(negative) definite. Definition 29 (p.165) . Let A be an m × m matrix. A is said to be nonsingular if there exists an m × m matrix A - 1 , called the inverse of A, such that A - 1 A = AA - 1 = I Theorem 45 (Thm 8.5, p.165) . If A has an inverse, it is unique. Theorem 46 (Thm 23.16, p.621) . Let A be a symmetric m × m matrix. Then there exists a nonsingular matrix P such that (i) P - 1 = P T , and (ii) P - 1 AP = P T AP = r 1 0 ··· 0 0 r 2 ··· 0 . . . . . . . . . . . . 0 0 ··· r m If we write the diagonal matrix as D , then we have A = PDP T . Hence, x T A x = x T PDP T x = y T D y , where y = P T x . A is positive(negative) definite D is positive(negative) definite. 15 Unconstrained Optimization 2 (ch. 17) 15.1 Second Order Conditions Sufficient Conditions Theorem 47 (Thm 17.2, p.399) . Let F : U R 1 be a C 2 function defined on an open set U R m . Suppose that DF ( x * ) = 0 . (Such x * is called a critical point of F.) (1) If D 2 F ( x * ) is negative definite, then x * is a strict local max of F; (2) If D 2 F ( x * ) is positive definite, then x * is a strict local min of F; 41
(3) If D 2 F ( x * ) is indefinite, then x * is neither a local max nor a local min of F.

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.

{[ snackBarMessage ]}

Page1 / 7

math-2011-note10 - Example 6 Suppose that we have a...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online