[試題] 100上 陳釗而 數量方法入門 期中考

[isolator (asdfzxcvqwer)]

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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