課程名稱︰應用數學一
課程性質︰物理系必帶
課程教師︰黃宇廷
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2019/06/18
考試時限（分鐘）：180
試題 :
1.
Reduce the equation 12u^2-12√2uv+18v^2 = 1 to a sum of square by finding the eigenvalues of the corresponding A, and sketch the ellipse.(20pts)
2.
If a matrix has eigenvectors (1,4,2) and (2,-1,1), with eigenvalues 4 and 5 respectively, please construct the matrix.(10pts)
3.
A matrix M is deifned on the basis vectors V_1 = (1,3) V_2 = (-1,5). Please construct a similarity transformation that would convert it to the basis U_1 = (2,5), U_2 = (1,4).(10pts)
4.
Find the eigenvalues and orthogonal eigenvectors of
A = 1 1 1 1 B = 1 0 1 0
1 1 1 1 0 1 0 1
1 1 1 1 1 0 1 0
1 1 1 1 0 1 0 1
(10pts)
5.
Given matrices A and B, in general we have exp(A)exp(B) = exp(A+B+[A,B]/2+...)
What is the next term in ... ?(20pts)
Hint:it is given by a sum of terms involving products of 3 matrices.
6.
Consider the integral
2 ∞
Π [∫dx_i ]exp(-<x|A|x>/2+B^T|x>)
i=1-∞
where A = (3 1) and B^T = (1,2) ans |x> = (x_1)
1 3 x_2
Please complete the integral.(20pts)
7.
Find A to change y''=5y'+4y into a vector equation for u(t) = (y(t),y'(t))
du/dt = Au
whar are the eigenvectors of A? Find it in another way by substituting y=e^{λt} into the differential equation.(10pts)
8.
Multi-national companies in the US,TW and Euro have assets of 4 trillion USD. In the beginning 2 trillion are in the US ans 2 trillion in Euro. Each year half of the US money stays and 1/4 goes to Euro and 1/4 goes to TW(yeah!). For TW and Euro 1/2 stays home and 1/2 goes to US.(15pts)
(a)
Find the matrix that gives
US US
( TW ) = A (TW )
Euro year k+1 Euro yeark
what is its eigenvalues?
(b)
How is the money distributed when the world ends?
(c)
Find the limiting distribution of money at year k.