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. ---------------------------------------------------------------------------- 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 . ---------------------------------------------------------------------------- 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? ---------------------------------------------------------------------------- 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 ---------------------------------------------------------------------------- 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 ┘ ---------------------------------------------------------------------------- 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. ---------------------------------------------------------------------------- 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)