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ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutat...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pi...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following...
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following steps (a)(b) to obtain the LU decomposition of the matrix A with partial piv- oting (a) Apply the Gaussian elimination method with partial pivoting to obtain an upper trian- gular matrix U. Record the corresponding permutation matrix for each pivoting step, and the numbers lik used to eliminate the zeros in column k. (b) Based on (a), express the matrices P, L and U...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can write it as the product A = LU of a lower triangular matrix and an upper triangular matrix. Show that the matrix 0 1 21 A= 3 4 5 (6 7 9] does not have an LU decomposition 0 0 Uji U12 U13 O 1 2 Il 21 l22 0 0 U22 U23 = 3 4 5 (131 132 133 0 0 U33 6...
numerical analysis question 3. For the matrix in problem 1, apply the Householder transformation to transform...
numerical analysis question
3. For the matrix in problem 1, apply the Householder transformation to transform A into an upper triangular matrix R. Express all the Householder transformation matrices p(1), p() explicitly. Based on the orthogonal matrices p(1), p(2) and the upper triangular matrix R, obtain the QR decomposition of the matrix A. A = | 1 4 il 2 3 2 -2 2 1
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6...
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6 -2 1 a. (4 pts) Find the multipliers l21, 131,132 and the elemention matrices E21, E31, E32 b. (2 pts) Use the multipliers l21, 131,132 to construct the lower triangular matrix, L c. (2 pts) Use the elimination matrices to determine the upper triangular, U, matrix of A d. (2 pts) verify that LU A
Let T be a linear operator on a finite dimensional vector space with a matrix representation...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
Function LUfac_solver.m is provided here: function [x] = LUfac_solver(LU,b,piv) % % function [x] = LUfac_solver(lu,b) %...
Function LUfac_solver.m is provided here:
function [x] = LUfac_solver(LU,b,piv)
%
% function [x] = LUfac_solver(lu,b)
%
% This program employs the LU factorization to solve the linear
system Ax=b.
%
% Input
% LU: lu matrix from GEpivot_new function
% b: right side column vector (ordered corresponding to original
vector
% sent to GEpivot_new)
% piv: vector indicating the pivoting (row interchanges that
took place
% during GE
%
% Output
% x: solution vector
%
% Written by Steve...
linear algebra Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and...
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the...
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following steps (a)(b) to obtain the LU decomposition of the matrix A with partial piv- oting (a) Apply the Gaussian elimination method with partial pivoting to obtain an upper trian- gular matrix U. Record the corresponding permutation matrix for each pivoting step, and the numbers lik used to eliminate the zeros in column k. (b) Based on (a), express the matrices P, L and U...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can write it as the product A = LU of a lower triangular matrix and an upper triangular matrix. Show that the matrix 0 1 21 A= 3 4 5 (6 7 9] does not have an LU decomposition 0 0 Uji U12 U13 O 1 2 Il 21 l22 0 0 U22 U23 = 3 4 5 (131 132 133 0 0 U33 6...
numerical analysis question
3. For the matrix in problem 1, apply the Householder transformation to transform A into an upper triangular matrix R. Express all the Householder transformation matrices p(1), p() explicitly. Based on the orthogonal matrices p(1), p(2) and the upper triangular matrix R, obtain the QR decomposition of the matrix A. A = | 1 4 il 2 3 2 -2 2 1
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6 -2 1 a. (4 pts) Find the multipliers l21, 131,132 and the elemention matrices E21, E31, E32 b. (2 pts) Use the multipliers l21, 131,132 to construct the lower triangular matrix, L c. (2 pts) Use the elimination matrices to determine the upper triangular, U, matrix of A d. (2 pts) verify that LU A
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
Function LUfac_solver.m is provided here:
function [x] = LUfac_solver(LU,b,piv)
%
% function [x] = LUfac_solver(lu,b)
%
% This program employs the LU factorization to solve the linear
system Ax=b.
%
% Input
% LU: lu matrix from GEpivot_new function
% b: right side column vector (ordered corresponding to original
vector
% sent to GEpivot_new)
% piv: vector indicating the pivoting (row interchanges that
took place
% during GE
%
% Output
% x: solution vector
%
% Written by Steve...
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
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