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28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 Algenra Quiz
2010/05/28
1. (30pt) Judge if the two matrices in each of the following sets are similar.
Explain your answer.
(a) (15pt)
┌ 2 1 0 0 ┐ ┌ 2 1 0 0 ┐
｜ 0 2 0 0 ｜ and ｜ 0 2 1 0 ｜
｜ 0 0 2 1 ｜ ｜ 0 0 2 1 ｜
└ 0 0 0 2 ┘ └ 0 0 0 2 ┘
(b) (15pt)
┌ 2 1 0 0 ┐ ┌ 2 0 0 0┐
｜ 0 2 1 0 ｜ and ｜ 1 2 0 0｜
｜ 0 0 2 1 ｜ ｜ 0 1 2 0｜
└ 0 0 0 2 ┘ └ 0 0 1 2┘
2. (40pt) For the following matrix A, find a real matrix S such that S^(-1)AS
is Jordan canonical form.
┌ 2 -1 1 1 ┐
A = ｜ 0 1 1 1 ｜
｜ 0 0 2 0 ｜
└ 0 -1 -3 -1 ┘
┌ 0 1 0┐
3. (30pt) Let A = ｜ -1 0 1｜，
└ 0 -1 0┘
(a) Find the eigenvalues and eigenvectors of A. If there are conjugate
complex eigenvalues, make their eigenvectors conjugate too.
(b) Use the above result to find a real matrix S such that S^(-1)AS is the
following matrix:
┌ 0 0 0 ┐
｜ 0 0 √2 ｜
└ 0 √2 0 ┘
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