11: Given the function f(x) = 8x² - 4x₁x₂ + 5x² and Xo = [5 2]. Then G¹g at Xo is equal (( G is the Hessian matrix and g is the gradient vector for f(X))) A: [25] B: [-2 -5] C: [5 2]T D: [-5 -2]T A B C D
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![11:
Given the function f(x) = 8x² - 4x₁x₂ + 5x² and Xo = [5 2]. Then G¹g at Xo is
equal (( G is the Hessian matrix and g is the gradient vector for f(X)))
A: [25]
B: [-2 -5] C: [5 2]T
D: [-5 -2]T
A
B
C
D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93d39ace-d211-4a4f-8c4a-ac236bd6fdac%2Ff28f38e1-eef9-4c82-bc0f-ed290aa033f2%2Fwk110mc_processed.jpeg&w=3840&q=75)
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- Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in a line and connected by springs with constants k12, k23, k34 whose natural lengths of extension are l12, l23, l34. Let f1, f2, f3, f4 denote the rightward forces on the masses, e.g., f1 = k12(x2 - x1 - l12). (a) Write the 4 x 4 matrix equation relating the column vectors f and x. Let K denote the matrix in this equation. (b) What are the dimensions of the entries of K in the physics sense (e.g., mass times tim, distance divided by mass, etc.)? (c) What are the dimensions of det(K), again in the physics sense? (d) Suppose K is given numerical values based on the unit meters, kilograms, and seconds. Now the system is rewritten with a matrix K' based on centimeters, grams, and seconds. What is the relationship of K' to K? What is the relationship of det(K') to det(K)?For a linear algebra matrix with initial value, how would I set up a general formula for xk to determine the limit as k approaches infinity for xk?For a 4×4 matrix whose top three rows are arbitrary and whose bottom row is (0, 0, 0, 1), show that the points (x, y, z, 1) and (hx, hy, hz, h) transform to the same point after homogenization.
- What should k be for the matrix given in the question to be invertible?The coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss-Seidel method converge anyway. Demonstrate that this is true of the Gauss-Seidel method, starting with the zero vector as the initial approximation and obtaining a solution that is accurate to within 0.01.Find all stationary points of the function f(x,y)= 3x2 -6xy+2y3, and use Hessian Matrix and the determinant to classify each as a local maximum, a local minimum, or a saddle point.
- Find all values of h for which the matrix B is not invertible.compute th e first four iterates, using th e zero vector as th e initial approximation, to show that th e Gauss-Seidel method diverges. Then show that the equations can be rearranged to give a strictly diagonally dominant coefficient matrix, and apply the Gauss-Seidel a - 4b + 2c = 2 ,2a + 4c = 1 , 6a -b- 2c = 1Generate a slope field in that shows the solution curve in the RF-plane (rabbit/fox) for the system of predator-prey equations with initial conditions R(0)=8 and F(0)=1