課程名稱︰管理數學
課程性質︰財金系大二必修
課程教師︰謝承熹
開課系所︰財金系
考試時間︰2006/01/09 14:30～16:30
試題 :
Ⅰ.(56 points) Answer each of the following as true (T) or false (F). Justify
your answer, or you cannot get any point. Moreover, please give your answers
in order. Thanks!
For problems 1 to 7, let Ａ be an n×n, idempotent, diagonalizable, and
non-zero matrix. Moreover, matrix Ａ is similar to a diagonal matrix Ｄ,
with Ｐ^(-1)ＡＰ=Ｄ, whose diagonal elements are the eigenvalues of Ａ,
while Ｐ is a matrix whose columns are respectively the n eigenvectors
of Ａ.
1. Ａ'Ａ is positive definite.
2. Ａ^(-1/2) exists.
3. Trace(Ａ)=Trace(Ｄ).
4. Trace(Ａ)=Rank(Ｄ).
5. You are given the following results: rank(ＡＰ)≦min{rank(Ａ), rank(Ｐ)},
then rank(Ａ)=rank(ＡＰ).
6. Rand(Ａ)=Rank(Ｄ).
7. The nullity of Ａ=0.
Ⅱ.(22 points) Let ｙ, Ｘ, β be an n×1 vector, an n×k matrix with n>k, and
an k×1 vector respectively. Moreover, answer Ｘ is full rank. Please find
β to minimum the objective function ｅ'ｅ, where ｅ=ｙ-Ｘβ.
(Note: You must check the second order condition to ensure that the β you
find really minimum the objective function.)
Ⅲ.(22 points) Let Ｖ be an n×n symmetry and positive definite matrix. Now
define Ａ=１'Ｖ^(-1)ｅ, Ｂ=ｅ'Ｖ^(-1)ｅ, Ｃ=ｅ'Ｖ^(-1)ｅ, where １ and ｅ
are two n×1 vectors.
(1) Simplify (Ａｅ－Ｂ１)'Ｖ^(-1)(Ａｅ－Ｂ１) in terms of Ａ, Ｂ, and Ｃ.
(2) Is (Ａｅ－Ｂ１)'Ｖ^(-1)(Ａｅ－Ｂ１) > 0 ? Why ?
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