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06月23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 Final
2010/06/23
A. [25%] Choose 5 problems to answer (choose only one form each set Xa./Xb.).
Judge if the statemnets are correct or not. Supply a simple explanation or
give a counterexample.
1a. If A is a m ×n non-zero matrix and m≧n, then (A^T)A is a positive
definite matrix.
1b. If A is positive definite, then there is always a matrix B such that
B^2 = A.
2a. Let A be a order n complex square matrix. Suppose A is Hermitian (i.e.
A^H = A or A^* = A), then det(A)∈|R.
2b. If the matrix A+iB is Hermitian, where A and B are real square matrices.
Then A and B are both symmetric matrices.
┌λ １ ０┐ ┌ λ^(-1) １ ０ ┐
3a. λ≠0. If A~｜０ λ １｜,then A^(-1) ~｜ ０ λ^(-1) １ ｜.
└０ ０ λ┘ └ ０ ０ λ^(-1)┘
3b. A is an order 4 square matrix. Suppose the characteristic polynomial
f_A(t) = (t-1)^4 and the minimal polynomial m_A(t) = (t-1)^2, then
┌ 1 1 0 0 ┐
A ~ ｜ 0 1 0 0 ｜
｜ 0 0 1 1 ｜
└ 0 0 0 1 ┘
4a. If the order n real square matrix A has eigenvalues λ1,...,λn, then the
singular vaules of A are |λ1|,...,|λn|.
4b. If the order n real square matrix A has singular vaules σ1,...,σn
(0 included), then the singular vaules of A^2 are (σ1)^2 , ...,(σn)^2.
5. Define a linear map T: R^n → C^n by T(e_i)=e_i, where e_i is the standard
basis for both R^n and C^n. Thus T is an linear isomorphism.
6. A: C^n → C^n is a linear transformation. im A^T is an A-invariant
subspaces of C^n.
7. Suppose A: R^n → R^n is a linear transformation and m≠n, then (A^+)A
cannot be I_n.
B. [75%] For each problem, wirte down detail computation or complete reasoning.
1. [15%] For the following Hermitian matrix A, find a unitary matrix U to
diagonalize A.
A = ┌ 2 1+i ┐
└ 1-i 2 ┘
2. [15%] For the following matrix A, find its first singular vaule
approximation and its pseudo-inverse.
┌ 0 1 0 ┐
A = ｜ 2 0 2 ｜
｜ 0 1 0 ｜
└ 2 0 2 ┘
3. [15%] For the following matrix A, find a matrix S such that S^(-1)AS is
the Jordan canonical form.
┌ 0 0 0 1 ┐
A = ｜ 1 0 0 -2 ｜
｜ 0 1 0 0 ｜
└ 0 0 1 2 ┘
4. [15%] Suppose the m ×n real matrix A has singular vaules σ1,σ2,...,σr,
where r = rank(A). Discuss the eignevaules and eignespaces of the
following matrix:
┌ 0 A ┐
└ A^T 0 ┘_(m+n) ×(m+n)
5. [15%] A real symmetric matrix A is positive definite if (v^T)Av > 0,
∀v≠0, prove that A is positive definite if and only if A satisfies the
Sylvester criterion
｜ a11 … a1i ｜
d_i = ｜ … … … ｜ > 0, i=1,2,...,n
｜ ai1 … aii ｜
C. Bonus. Solve no more than one problem to increase your credit.
1. [5%] By using SCD, prove that all invertible real matrix A has polar
decomposition A = QH, where Q is orthogonal and H is positive definite.
2. [5%] A is an order n square matrix. Prove that A ~ A^T. (Assuming Jordan
Form Theorem)
3. [10%] Let A be an order n real square matrix with singular vaules σ1≧σ2
≧…≧σn (0 included). If λ is a real eigenvaule of A, prove that
σn≦|λ|≦σ1.
4. [10%] Let A be an order 3 real square matrix. Consdier A as a linear
tragnsformation A : |R^3 → |R^3. Discuss the imgae of the unite sphere
x^2 + y^2 + z^2 = 1 under the map A.
5. [10%] Prove that
┌ λ a12 a13 … a1,(n-1) a1n ┐ ┌λ １ ０ … ０ ０┐
｜ ０ λ a23 … a2,(n-1) a2n ｜ ｜０ λ １ … ０ ０｜
｜ ０ ０ λ … a3,(n-1) a3n ｜～｜０ ０ λ … ０ ０｜
｜ … … … … … … ｜ ｜… … … … … …｜
｜ ０ ０ ０ … λ a(n-1)n｜ ｜０ ０ ０ … λ １｜
└ ０ ０ ０ … ０ λ ┘ └０ ０ ０ … ０ λ┘
⇔ a_(i,i+1)≠0, i=1,2,...,n-1
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