課程名稱︰應用數學一
課程性質︰系必修
課程教師︰陳義裕
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2008/06/10
考試時限（分鐘）：180分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(20 points)
(a)(6 points) Briefly describe what Gram-Schmidt orthogonalization process
for a finite-dimensional inner product space is.
(b)(6 points) The following list of matrices are linear operators expressed
in some chosen orthonormal basis. Please identify (no need to explain) all
the self-adjoint operator(s), if any. (The inner product can be
complex-valued.)
┌ ┐ ┌ ┐ ┌ ┐
│ 1 2i 4∣ │sqrt(3) 2 1│ │ 1 2 4│
A1 ≡│ 2i -2 5∣ A2 ≡│ 2 -2i -4│ A3 ≡│ 7 -2 7│
│ 4 5 3∣ │ 1 -4 3│ │ 4 2 1│
└ ┘ └ ┘ └ ┘
┌ ┐ ┌ ┐ ┌ ┐
│ 5 -3 2∣ │ 2 1 -3i│ │ 2 i 7│
A4 ≡│-3 2 1∣ A5 ≡│ 1 sqrt(3) 5 │ A5 ≡│ 5 3 -i│
│ 2 1 7│ │ 3i 5 1 │ │ 1 5 2│
└ ┘ └ ┘ └ ┘
(c)(8 points) Can the following be the matrix representation of a
self-adjoint operator in some as yet unspecified basis of R^2 with
standard inner product?
┌ ┐
│ 1 1 │
│ 4 1 │
└ ┘
Please note that no credits will be granted unless you give the correct
answer and also support your claim with a correct explanation.
2.(20 points) This problem deals with determinants.
(a)(10 points) Please compute the determinant of the following matrix:
┌ ┐
│ 1 2 1 3 1 │
│ 1 1 3 2 4 │
│ 2 2 -1 3 0 │
│ 3 0 -2 1 2 │
│ 2 3 -1 2 -1 │
└ ┘
Please note that you won't get any partial credits if you do not obtain
the correct answer, which should be a small integer.
(b)(5 points) Suppose we have an n × n real-valued matrix Aij which is
known to satisfy
ｎ
ΣAki*Akj = δij*(aj)^2 (no sum on j)
k＝1
for some positive number aj. Here, δij is the Kronecker delta function
satisfying
1 if i ＝ j
δij = {
0 if i ≠ j
Please show that
︿
│det(Ａ)│ = a1 * a2 * … * an
(Note: It is almost a one-line proof if you use certain properties of
determinants we showed in the class.)
(c)(5 points) Please give a geometrical interpretation of the result of (b).
︿
3.(30 points) Let Ｌ: U → U be a linear operator acting on an n-dimensional
︿
complex-valued inner product space U. It is known that Ｌ is a unitary
︿ ︿+︿ ︿︿+ ︿ ︿
operator, i.e., Ｌ satisfies Ｌ Ｌ＝ＬＬ ＝Ｉ, where Ｉ is the identity
︿ ︿
operator on U and Ｌ is the adjoint operator of Ｌ.
→ ︿ ︿ →
(a)(5 points) Suppose ｕ is an eigenvector of Ｌ, i.e., Ｌ(ｕ)＝λｕ for
→ →
some number λ, and ｕ≠ 0. Let W be the orthogonal complement to ｕ.
︿+
Please show that W is an invariant subspace of Ｌ , that is,
︿+
Ｌ (W) ○ W.
(b)(5 points) Please use the fact of (a) (which says that W is an invariant
︿ ︿
subspace of Ｌ) to prove that W is also an invariant subspace of Ｌ.
(Note: You can simply quote the result of (a) for this problem even if you
do not know how to prove (a).)
(c)(10 points) Please prove that we can find an orthonormal basis
→ → → ︿ ︿ → →
{e1, e2, ..., en} of U such that Ｌ is diagonal, that is L(ej) ＝ λj*ej
for some number λj.
→ ︿ →
(d)(5 points) Suppose ｕ is an eigenvector of Ｌ, please prove that ｕ is
︿+
necessarily an eigenvector of Ｌ , too
(e)(5 points) Please prove that the absolute value of any eigenvalue of a
unitary operator is always unity.
4.(30 points) A particle can be in one of two states: α or β. If it is
initially in state α, than at the next moment it has 1/4 probability to
remain in state α and 3/4 probability to "jump" to state β. But if it is
initially in state β, then at the next moment it has equal probability
(＝1/2) to be in state α or β.
(a)(5 points) Let pαn and pβn denote the probability of the particle in
state α and β at the n-th moment, respectively. Define
┌ ┐ ┌ ┐
→ │ pαn │ ︿ │ 1/4 1/2 │
rn ≡│ │ and Ａ≡│ │
│ pβn │ │ 3/4 1/2 │
└ ┘ └ ┘
Please explain why we have
→ ︿ →
ｒn+1 ＝ Ａ(rn).
(b)(10 points) Please find all the eigenvalues and their associates
︿
eigenvectors of the matrix Ａ.
(c)(10 points) Please give a rigorous argument showing that we will end up
having same probabilistic distribution when the "time" n goes to infinity,
irrespective or whatever initial distribution is given.
(d)(5 points) What is this final "equilibrium" probabilistic distriion?
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