2. Consider a one-factor model:Y = BF +Uwhere Y is the data N x T dimensional; B is N x 1 dimensional vector ofbetas; F is 1 × T dimensional vector of factors. We estimate B and F usingthe following method:min ||Y – BF||B,Fwith constraint1F*F= 1Show that the solution F//T is the eigenvector of the T ×T matrix Y'Y.

Given information:The question presents a one-factor model represented by the equation , where is…

Answered: 2. Consider a one-factor model: Y = BF…

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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Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4, 4)} be bases for R², and let 23 = [33] 04 A = R2 relative to B. (a) Find the transition matrix P from B' to B. be the matrix for T: R² ->>> P = 6 9 [V] B [T(V)]B = (b) Use the matrices P and A to find [v] and [T(v)]B, where [V] B¹ = [-4 3]. -12 -1/3 -24 -96 4 -96 4 11 ← (c) Find P-1 and A' (the matrix for T relative to B'). 1/3
7. Let A be an invertible n x n matrix over R. Consider the system of linear equation Ax = b or 2aija; = bị, AijX'j i = 1,..., n. Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(k) +b, k = 0, 1,2,.. . Let (3) :). 4 0 0 4 0 0 0 4 (3) 4 -1 -1 4 -2 A = -1 C = b = 2 *(0) 4 2 The iteration converges if p(C-!R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-!R)< 1. Perform the iteration.
Use the Finite Difference method to write the equation x" + 2x' - 6x = 2, with the boundary conditions x(0) = 0 and x(9)-0 to a matrix form. Use the CD for the second order differences and the FW for the first order differences with a mesh h=3.
Suppose a, = (1 -1 1 1) 5. az = (1 0 1 0) %3D %3D az = (1 1 1 1)''a¸ = (3 2 3 0)" - Please find out whether a1,a2,a3,&4 are linear independent or not?
a' = with one of the phase plane direction fields below: A B D (The blue lines are the arrow shafts, and the black dots are the arrow tipa.) Note: To solve this problem, you only need to compute eiguvalus. In fact, it is enough to just compute whether the cigenvalues are real or complex and positive or negative, If you use Trace-Determinant plane, give a brief explanation about eigenvalues.
dx Solve the initial value problem xdy + dy= ydx, yca) =6 Given dt- sinx-cOSX + 1, fC0)=4 = Sinx-COSX +1, fC0)=4 dx find the function fox) quare matriX A is invertible Fo line Solutior
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