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課程名稱︰線性代數二
課程性質︰必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012/4/14
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
There are problems A to F with a total of 140 points. Please write down your
computational or proof steps clearly on the answer sheets.
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A. Consider the linear ODE system du/dt = Au, where A = ┌ 0 -1 0 ┐.
│ │
(a) (5 points) Without solving the problem, prove │ 1 0 -1 │
│ │
that || u(t) || is a constant. └ 0 1 0 ┘
2
A
(b) (5 points) Prove that e is orthogonal.
tA
(c) (15 points) Compute e .
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2 2 2
B. Define the quadratic form Q(x) = x + 5x + 9x + 4x x + 6x x + 8x x
1 2 3 1 2 1 3 2 3
3 3
in R , where x = [ x x x ] ∈ R .
1 2 3
(a) (2 points) Write down the symmetric matrix A ∈ M (3, R) so that
T
Q(x) = x Ax.
(b) (10 points) Find a matrix P ∈ GL (3, R) such that Q(x) =
3 2
Σ λ y for some constants λ , λ , λ ,
j=1 j j 1 2 3
T
where [ y y y ] = Px. What is the inertial index
1 2 3
of A? Prove that if P is orthogonal, then λ , λ , λ
1 2 3
are eigenvalues of A.
(c) (3 points) Is there a real solution to the equation Q(x) = -1?
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C. Given the matrix.
A = ┌ 1 1 ┐ (a) (10 points) Find the singular values, and the
│ │
│ 1 0 │ SVD-decomposition of A.
│ │ +
└ 0 1 ┘ (b) (6 points) Use (a) to find the pseudo-inverse A ,
and the minimal least square approximate
T
solution to Ax = [ 1 3 0 ] .
(c) (4 points) Evaluate min{||A-B||, B ∈ M (3 ×2, R) and B has rank 1}.
2
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D. (20 points) Apply Householder matrices to reduce the matrix ┌ 1 2 4 ┐
│ │
to upper Hessenberg form, and also find its │ 2 6 7 │
│ │
QR-decomposition. └-2 1 8 ┘
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E. Prove the following statements. Each has 10 points.
(a) Let B ∈ M (n, R) has rank r. Prove that the symmetric matrix
A = ┌ I B ┐ ∈ M (2n, R) has n positive eigenvalues and r
│ n │
│ T │ negative eigenvalues.
└ B 0 ┘
(b) The two Hermitian matrices A ∈ M (n+1, C) and B ∈ M (n, C) are
n
related by A = ┌ B v ┐, where v ∈ C and α ∈ R.
│ │
└ v* α┘
If σ(A) = {λ >= λ >= ... >= λ } and
1 2 n+1
σ(B) = {β >= β >= ... >= β }, prove that
1 2 n+1
λ >= β >= λ >= β >= ... >= λ >= β >= λ .
1 1 2 2 n n n+1
(c) Suppose that two positive definite matrices P, Q ∈ M (n, F)
2 2
satisfy P = Q , prove that P = Q.
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F. Determine which of the following statements is true. Prove your answer.
Each has 6 points.
A
(a) The exponential e for A ∈ M (2, R) maps M (2, R) onto to the set
+
GL (2, R) = {B ∈ GL (2, R) | det(B) > 0}.
(b) If det(A) = 1 for A ∈ M(n, R), then the linear system Ax = b is a
n
well-conditioned problem with respect to the Euclidean norm of R .
(c) Let A ∈ M (n, F). If A + A* is negative defininte, then A is a
stable matrix.
X
(d) Since exponentials of real numbers are positive, the equation e = -I
2
has no solutions in M (2, R).
(e) Let A ∈ M (n, F) be a positive definite matrix. Then for any matrix
B ∈ M (n, F), the matrix AB is similar to a congruent matrix of B.
※ 編輯: arsenefrog 來自: 140.112.239.2 (04/17 01:52)