课程名称︰工程数学上 课程性质︰系定必修 课程教师︰杨馥菱 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰980112 考试时限（分钟）：110 是否需发放奖励金：是 试题 : Write down the important steps of your calculations. If there exist any applicable theorem that simplifies the calculations, state the theorem to support your quick answer. Problem 1. (35%) Orthogonality of eigenspace ┌４ -1 １┐ Consider A = │-1 ４ -1│ └１ -1 ４┘ (1) Determine the characteristic polynomial of A and find the eigenvalues λi. (Number λi such that λ1≧λ2≧λ3). Determine the corresponding eigenvector Ei. (2) Are the eigenvectors E1, E2, E3, linearly independent to each other? (3) Use the eigenvictors to form a transformation matrix P = [E1 E2 E3] such -1 that P AP = Λ. What is Λ? (4) Are {E1, E2, E3} orthogonal to each other? If ehry are not, try to find a set of orthogonal eigenvectors {G1, G2, G3}. (Hint: Use the additional condition that Gi‧Gj = 0 (i≠j). (5) Use Gi's to develop a set of orthonormal eigenvectors (G*1, G*2, G*3}. Form a new transformation matrix Q = [G*1 G*2 G*3]. -1 -1 (6) Determine Q and Q AQ. Problem 2. (20%) Cayley-Hamilton Theorem for diagonalizable square matrix ┌1 0 0┐ Given A = │0 1 1│ └0 1 1┘ (1) Is A diagonalizable? Give reasons. 3 2 (2) Find the characteristic polynomial p(λ) = λ + aλ + bλ + c = 0 for A. 3 2 (3) Show that p(A) = 0. In other word, show A + aA + bA + cI = 0. 4 (4) Use(3) to calculate A ? -1 (5) Similarly, compute A . Problem 3. (30%) System of ODEs . dxi Consider a system of inhomogeneous 1st order ODEs (where xi = ---) . t dｔ x1 = x1 + 2e ┌ 2 ┐ { . subjected to: x(0) = │ │ x2 = -x1 + 3x2 + sin t └1/2┘ . (1) Present the problem in a matrix form: x = Ax + B(t) (2) Find the eigenpairs of A to form the fundamental matrix Ω(t) for the problem. -1 (3) Determine Ω (t). (4) Calculate the particular solution Xp(t). (5) Present the general solution as x(t) = Ω(t)C + Xp(t). Determine the constant matrix X with the initial condition. -at -at t -at 1-e cos(t) - ae sin(t) (Hint: Can apply ∫ e sin(τ) dτ = ---------------------------) 0 2 1 + a Problem 4. (25%) System of difference equation The following set of equations state how the values of x and y at next iteration (the n+1 step) depend on the value at the previous step (step n). 3 1 Xn+1 = -Xn - -Yn 4 4 ┌X0┐ ┌0┐ { with │ │ = │ │. -1 3 └Y0┘ └1┘ Yn+1 = ─Xn + -Yn ４ 4 ┌Xn+1┐ ┌Xn┐ (1) Use Zn+1 = │ │, Zn = │ │ to rewrite the system of equations info └Yn+1┘ └Yn┘ Zn+1 = AZn. (2) Find the eigenvalues of (λ1≧λ2) and the associated eigenvectors E1, E2 for A. n n (3) Determine A E1 and A E2. (4) Represent the initial condition by E1, E2, i.e. obtain constants a,b such that Z0 = aE1 + bE2. n (5) Show that Zn = A Z0. Then use (3)~(5) to estimate lim Xn and lim Yn. n→∞ n→∞ -- As an Engineer, I shall participate in none but honest enterprises. When needed, my skill and knowledge shall be given without reservation for the public good. In the performance of duty and in fidelity to my profession, I shall give the utmost. excerpt from: The Obligation of an Engineer dn890221 --

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