课程名称︰应用数学一 课程性质︰系必修 课程教师︰陈义裕 开课学院：理学院 开课系所︰物理学系 考试日期（年月日）︰2008/06/10 考试时限（分钟）：180分 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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? --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.242.66 ※ PHONm:转录至看板 NTUscuba 11/28 14:41