WDocument1 - Microsoft Word (Product Activation Failed)FileHomeInsertPage LayoutReferencesMailingsReviewViewa ?% CutA Find -Calibri (Body)- 14-A AAaAalAaBbCcDc AaBbCcD AaBbC AaBbCc AaBI AqBbCclE Copyag ReplaceBI UabAI NormalPaste- *abe x, xAI No Spaci.. Heading 1Heading 2TitleSubtitleChangeWFormat PainterStyles - Select -ClipboardFontParagraphStylesEditing2:1·1•I 11• 2:1:3:1·4: 15.1 6:1 7:18•1 9.1 10. 11· I·12: 1 •13. 1 14: I 15. 1 : 17: 1 18.ORDINARY DIFFERENTIAL EQUATIONSPLEASE ANSWER ALL QUESTIONS(a)Show from first principles, i.e., by using the definition of linear inde-pendence, that if µ = x + iy, y # 0 is an eigenvalue of a real matrixA with associated eigenvector v = u+ iw, then the two real solutionsY(t) = eat (u cos bt – w sin bt)andZ(t) = e"(u sin bt + w cos bt)are linearly independent solutions of X = AX.(b)Use (a) to solve the systemX =-8X.NB: Real solutions are required.07:14 PM2022-05-18Page: 1 of 1Words: 9昆昌 90%+

Answered: Image /qna-images/answer/0cbe8617-11ce-4ce4-aed1-df12240aee31.jpg

Answered: (a) Show from first principles, i.e.,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The options for matching the values of c1 and c2 are -4 through 1.
2. Consider a one-factor model: Y = BF +U where Y is the data N x T dimensional; B is N x 1 dimensional vector of betas; F is 1 × T dimensional vector of factors. We estimate B and F using the following method: min ||Y – BF|| B,F with constraint 1 F*F= 1 Show that the solution F//T is the eigenvector of the T ×T matrix Y'Y.
Apply the eigenvalue method to find a general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. x', = - 2x, + 8x2. x'2 = 12x, - 6x2 What is the general solution in matrix form? x(t) =O
With two independent eigenvectors?
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 3 + 5i and corresponding eigenvector i * = (-²+1). v= Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Aỹ, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. (t) = 2(t) = → - →
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
Suppose A is a 2 x 2 real matrix with an eigenvalue λ = 2 + 4i and corresponding eigenvector −1+ *-[-'+']. = Determine a fundamental set (i.e., linearly independent set) of solutions for y' = Ay, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. y₁(t) = 19 y₂(t) =
Suppose the eigenvalues associated with a 2 by 2 matrix of a system of linear ODE's are complex conjugates. (a) Use Euler's formula to rewrite the solution in the form x(t) = u(t) + iv(t). (b) Show that x(t) = u(t) + v(t) is also a solution of the system.
For the following matrix: a) find the eigenvalues and the eigenvectors b) check the answer by using the orthogonal property (x(i))T(x(j))=1(i≠l) c) determine the three normalized eigenvectors by using (x(i))T(x(i))=1(i≠l)
Let M be a 2 x 2 matrix with eigenvalues A₁ = 0.3, A2 = -0.25 with corresponding eigenvectors = [2²] ²2² = [9] · V2 Consider the difference equation with initial condition xo That is, write xo = C1V1 + C₂V2 H Write the initial condition as a linear combination of the eigenvectors of M. In general, X = Specifically, X4 V1 = For large k, xk Xk+1 = V₁+ )k v₁+ Mxk A V2 7)k V₂
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
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