課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰翁秉仁
開課學院：理學院
開課系所︰數學系
考試日期︰2010年05月04日(五)，11:20-12:10
考試時限：50分鐘
是否需發放獎勵金：是
試題 :
Linear Algebra Quiz
2010/05/14
1.(30pt)Find the range of x such that the follwoing matrix is postive definite.
┌ x 1 0 ┐
｜ 1 x -1 ｜
└ 0 -1 x ┘
2. (30pt) Whcih one of the following matrices is positive definite. Explain
your answer. Directly apply Sylvester's criterion is NOT encouraged.
┌ 1/7 1/6 1/5 1/4 ┐ ┌ 1 -1/2 0 -1/2 ┐ ┌ 6 1 1 1 ┐
｜ 1/6 1/5 1/4 1/3 ｜ ｜ -1/2 1 -1/2 0 ｜ │ 1 5 1 1 │
｜ 1/5 1/4 1/3 1/2 ｜ ｜ 0 -1/2 1 -1/2 ｜ │ 1 1 0 1 │
└ 1/4 1/3 1/2 1 ┘ └ -1/2 0 -1/2 1 ┘ └ 1 1 1 1 ┘
┌ 10 -1 -2 -4 ┐
｜ -1 10 -6 -3 ｜
｜ -2 -6 10 -2 ｜
└ -4 -3 -2 1 ┘
3. (40pt) A Cholevshy decomposition of a positive definite matrix A is A=(B^T)B
,such that B is upper triangular with positive diagonal entries.
(a) (30pt) Find the Cholevsky decomposition of
┌ 2 -1 0 ┐
D3 = ｜ -1 2 -1 ｜
└ 0 -1 2 ┘
(b) (10pt) Guess a general rule for the Cholevsky decomposition of the
follwing matrix and verify it.
┌ 2 -1 0 … 0 0 ┐
｜ -1 2 -1 … 0 0 ｜
｜ 0 -1 2 … 0 0 ｜
Dn = ｜ … … … … … … ｜
｜ 0 0 0 … 2 -1 ｜
└ 0 0 0 … -1 2 ┘
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