課程名稱︰工程數學上
課程性質︰系定必修
課程教師︰楊馥菱
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰980112
考試時限（分鐘）：110
是否需發放獎勵金：是
試題 :
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→∞
--
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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
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