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. --

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