16.1 For each of the matrices = [3 2] A = and B= 1 0 -2 1 -2 1 0 1 -3 perform two iterations of the power method by hand starting with a vector of all ones. State the resulting approximations of the eigenvalue and eigenvector.
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code using Matlab pls
![16.1 For each of the matrices
= [3 2]
A =
and B=
-2 1 0
1 -2 1
0 1 -3
perform two iterations of the power method by hand starting with a vector of all ones. State the
resulting approximations of the eigenvalue and eigenvector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a8ab243-eb11-4f1d-bd98-efff8fce9d94%2F90454174-87e4-4a70-9f58-5fc9351ce57b%2Fx1r9i47_processed.jpeg&w=3840&q=75)
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- 3 : Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this i=1 purpose, compute the Hessian matrix of the function f(x).Q4: Write the parametric equation of revolution surface in matrix form only which generated by rotate a Bezier curve defined by the coefficient parameter in one plane only, for the x-axis [0,5, 10,4], y-axis [1,4,2,2] respectively, for u-0.5 and 0 = 45° Note: [the rotation about y-axis].2. Matrix Quantum Mechanics. Use python matrix and algebraic tools for this problem. Consider orthogonal vectors [1], [2) and [3). The operators C and P are represented by C = /C₁ 00 0 C₂ 0 0 0 C3. /P₁ 0 0 P = 0 0 P2 0 P3 Where all quantities are real. a) Do the operators C and P commute? b) Find the eigenvalues and normalized eigenvectors of the given operators.
- Generate H (Hilbert Matrix) for n = 15.Compute the right-hand size vector b = Hx so that the exact solution is xˆ = 1 (thevector of ones). Solve the linear system in MATLAB as a least squares problem usingSVD and report the relative error in the approximate solution in the Euclidean norm.Compare your result to the solution obtained by H\b.Use MATLAB to create a program for Gaussian Elimination.Important inclusions of the program:a. c0efficient and c0nstant matrices are separated.b. it can work on any size of matrixc. outputs the augmented matrix of the c0efficient and c0nstant matrices.d. outputs the R0w echel0n forme. outputs the soluti0n of systems of linear equati0ns using back substituti0nf. outputs the Reduced R0w echel0n form to verify answer in letter e.Consider the following. -4 2 0 1 -3 A = 0 4 2 2 1 2 2 (a) Verify that A is diagonalizable by computing P AP. P-lAP = E (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n xn matrices, then they have the same eigenvalues. VA
- Consider the following. 0 1 -3 A = 0 4 0 4 1 2 2 2 -1 2 (a) Verify that A is diagonalizable by computing P-AP. p-1AP = (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues.Let v be a vector whose coordinates are given as v = [vx, Vy, Vz. If the quaternion Q represents a rotation, show that the new, rotated coordinates of v are given by Q(0, Vx, Vy, Vz)Q*, where (0, vx, Vy, Vz) is a quaternion with zero as its real component.Perform the following Matrix Operations for the predefined matrices. Given the System of equations: 2х + 4y — 5z + Зw %3D —33 3х + 5у—2z + бw %3D — 37 х — 2у + 4z — 2w 3 25 Зх + 5у-3z + Зw = -28 Write the systems as Ax = b, where A is the coefficient matrix and b is the vector for the constants. 1. Encode the Matrix A and the column vector b. 2. Solve for Determinant of A. 3. Find the Inverse of A. 4. Form the Reduced Row Echelon of A. 5. Find the number of rows and number of columns of Ab. 6. Find the sum of the columns of A. 7. In each of the columns of A, find the highest values and its indices. 8. Augment A with b; 9. Find b\A 10. Form the Reduced Row Echelon of Ab. 11. Extract the Last Column of the Reduced Row Echelon Form of Ab. 12. Create a matrix A whose elements are the same as matrix A, but the first column is the column vector b. 13. Create a matrix A whose elements are the same as matrix A, but the second column is the column vector b. 14. Create a matrix A whose elements…
- ) Assume A is k × n-matrix and Q is n x n-invertible matrix. Prove that rank(AQ) = rank(A).. The determinant of an n X n matrix can be used in solving systems of linear equations, as well as for other purposes. The determinant of A can be defined in terms of minors and cofactors. The minor of element aj is the determinant of the (n – 1) X (n – 1) matrix obtained from A by crossing out the elements in row i and column j; denote this minor by Mj. The cofactor of element aj, denoted by Cj. is defined by Cy = (-1y**Mg The determinant of A is computed by multiplying all the elements in some fixed row of A by their respective cofactors and summing the results. For example, if the first row is used, then the determi- nant of A is given by Σ (α(CI) k=1 Write a program that, when given n and the entries in an n Xn array A as input, computes the deter- minant of A. Use a recursive algorithm.Write a MATLAB code to find the eigenvalues of the matrix A = [1 3 -1; 3 2 4; -1 4 -10] using the Jacobi method. (Full code with output)