課程名稱︰數量方法入門 課程性質︰必修 課程教師︰陳釗而 開課學院：社會科學院 開課系所︰經濟所 考試日期（年月日）︰100.09.02 考試時限（分鐘）：90 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. For any x,y∈R^n, show that ║x+y║<= ║x║+║y║. 2. Briefly explain the Projection Theorem (in mathematics or in words). 3. Let X be a matrix of order n*k and rank k. (a)Verify that M = I-Px is an orthogonal projection matrix, -1 where Px = X(X'X)X'. (b)Verify that MX = 0. (c)Rank(M) = ? 4. (Eigenvalues, idempotent matrices, and symmetry) (a)Show that the eigenvalue of an idempotent matrix are 0 and 1. (b)Show that if A is symmetric and has only eigenvalues 0 and 1, it is idempotent. 5. Show that if a matrix P is symmetric and idempotent, then P is positive semi-definite. -1 6. Let A be a nonsingular matrix. Consider the partition of A and matrix A -1 [ -1 -1 -1 ] A = [ W (-W)(A11)(A22) ] [ -1 -1 -1 -1 -1 -1 ] [ (-A22)(A21)W (A22)+(A22)(A21)(W)(A12)(A22) ] where A = [ A11 A12 ] [ A21 A22 ] is a partition of A such that A11 and A22 are square matrices, and we define -1 W = A11-(A12)(A22)(A21). Verify that for the linear regression model y = X1*β1 + X2β2 + ε the least squares estimator for β1 is given by ^ -1 -1 β1 = (X1'(I-Px2)X1)X1'(I-Px2)y, where Px2 = X2(X2'X2)X2'. ^ ^ ^ -1 Hint: X = [X1 X2], [β1 β2]' = β = (X'X)X'y, where X is a matrix of order n*k, X1 is a matrix of order n*k1, X2 is a matrix of order n*k2, β is an k*1 vector, β1 is an k1*1 vector, β2 is an k2*1 vector, and k1+k2 = k. 7. Let X be a matrix of order n*k and rank k, and let V be symmetric and positive difinite of order n. (a)Show that the matrix -1-1 Q:= I-V^(-1/2)X(X'VX)X'V^(-1/2) is symmetric and idempotent. (b)Use part (a) and the fact that X'V^(1/2)QV^(1/2)X = 0 to show that -1-1 -1 -1 (X'VX)X'V = (X'X)X' --

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