課程名稱︰線性代數二
課程性質︰必修
課程教師︰林紹雄
開課系所︰數學系
考試時間︰2006/6/16 10:20~12:50
Math 201 14410-01 (Linear Algebra) Final Exam
There are Problems A to E with a total of 110 points. Please write down
your computational, or proof steps clearly on the answer sheets.
A. (10 points) Use the Householder matrices to find the QR-decomposition of
the following matrix
[ 1 3 3 2 ]
[ 2 -4 1 1 ]
[ 2 -5 -1 -2 ]
B. (20 points) Find the singular values, the SVD-decomposition, and the
+
pseudo-inverse A of the matrix
[ 2 0 1 ]
[ 0 2 0 ]
Moreover, find the minimal length least square solution to Ax = b,where
b = [ 3 ]. Is b in the column space of A?
[ 0 ]
C. (10 points) Use the simplex method to minimize 2x_1+x_2 subjected to
x_1 + x_2 ≧ 4, x_1 + 3x_2 ≧ 12m x_1 - x_2 ≧, x_1 ≧ 0, x_2 ≧ 0.
D. Prove the following statements.
(a) (10 points) Let A and B be two matrices of order m*n and n*m
+
respectively. Prove that B = A iff (1) ABA = A, (2) BAB =B,
and (3) both AB and BA are symmetric.
(b) (20 points) Let A be a 2*2 positive definite matrix. Let J and L_ω be
the Jacobi and the SOR matrix of A (where ω is a real parameter). Let
ρ(L_ω) be the spectral radius of L_ω.
(1) Prove that ρ(L_ω) ≧ 1 for ω ≦ 0 or ω ≧ 2, and ρ(L_ω) ＜ 1
for 0 ＜ ω ＜ 2.
(2) Prove that the optimal SOR parameter ω_opt is given by
2
ω_opt = ────────────
1 + √(1 - (λ_1)^2)
(c) (10 points) Let A adn M be two n*n matrices which are symmetric, and
positive definite respectively. Prove that the smallest eigenvalue λ_1
Ax = λMx is not larger than the ratio a_11/m_11 of the corner entires.
E. Determine which of the following statements is true. Prove your answer (or
given a counterexample). Each has 10 points.
~
(a) Let A be a 2*2 real matrix. The shifted QR algorithm from A to A is
defined as: ~
A - a_22 I_2 = QR and A = RQ + a_22 I_2
where a_22 is the (2,2)-element of A. Now define A_0 = A, and
~
A_(n+1) = A_n for n = 1,2,.... Suppose that A has distinct real
eigenvalues, then A_n always converges to an upper triangular matrix as
n → ∞.
(b) Let B be an n*n invertible real matrix. Define the 2n*2n matrix A by
A = [ I_n B ]
[ B^T 0 ]
Then A is an indefinite matrix with n as its inertial index, and
det(A) = (-1)^n(det B)^2.
(c) If m*n matrix A has independent row vectors, the A has positive
+ T T -1
singular values and A = A (AA ) .
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