Math Review. ECON 510 Chapter 2 Optimization See Sydsaeter(2005, Chapters 2, 3)and Chiang(1984, Chapters 9, 11, 12 and 21) Positive definite matrix Definite matrices are directly related to optimization. A symmetric matrix A is positive semi- definite(A≥0) if rAr≥0,Vx; positive definite(A>0) if 2 Ar>0,Vx≠0: negative semi- definite(A≤0)ifx'Ax≤0,vx; negative definite(A< O) if xar<0,x≠0 Example 2.1.A= Example 2.2. Consider A Example23. For a symmetric matrix A,ifA≥0, then a≥0 for all i;ifA≤0, then a<0 for all i.■Chapter 2 Optimization Math Review, ECON 510 See Sydsæter (2005, Chapters 2, 3) and Chiang (1984, Chapters 9, 11, 12 and 21). 1. Positive Definite Matrix Definite matrices are directly related to optimization. A symmetric matrix A is positive semi-definite (A ≥ 0) if x0 Ax ≥ 0, ∀ x; positive definite (A > 0) if x0 Ax > 0, ∀ x 9= 0; negative semi-definite (A ≤ 0) if x0 Ax ≤ 0, ∀ x; negative definite (A < 0) if x0 Ax < 0, ∀ x 9= 0. Example 2.1. A = ⎛ ⎜⎝ 1 −1 −1 1 ⎞ ⎟⎠ ≥ 0.  Example 2.2. Consider A = ⎛ ⎜⎝ a b b c ⎞ ⎟⎠ .  Example 2.3. For a symmetric matrix A, if A ≥ 0, then aii ≥ 0 for all i; if A ≤ 0, then aii ≤ 0 for all i.  2—1