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Unformatted text preview: 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 = 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 = 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 < for some other x in R m . • Q ( x , y ) =- 3 x 2- 6 y 2 ≤ 0 and strictly negative except at . Hence, it is negative definite, and ( x , y ) = 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 = 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 (ii) P- 1 = P T , and (iii) P- 1 AP = P T AP = r 1 ··· r 2 ··· . . . . . . . . . . . . ··· 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 ··· r 2 ··· . . . . . . . . . . . . ··· r m • If we write the diagonal matrix as D , then we have A = PDP T ....
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This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.
- Spring '11