Homework Answers
MATLAB Script:
close all
clear
clc
fprintf('Part
1\n---------------------------------------\n')
A = [1 2 3 0 0 0 0 0;
2 1 2 3 0 0 0 0;
3 2 1 2 3 0 0 0;
0 3 2 1 2 3 0 0;
0 0 3 2 1 2 3 0;
0 0 0 3 2 1 2 3;
0 0 0 0 3 2 1 2;
0 0 0 0 0 3 2 1];
detA = det(A);
fprintf('det(A) = %-20.2f\n', detA)
fprintf('\nPart
2\n---------------------------------------\n')
invA = inv(A);
disp('inv(A) ='), disp(invA)
fprintf('\nPart
3\n---------------------------------------\n')
b = [2 4 8 16 32 61 128 256]';
x = A\b;
disp('Solution, x ='), disp(x)
fprintf('\nPart
4\n---------------------------------------\n')
x10 = evec(A, b, 10);
x100 = evec(A, b, 100);
disp('Approximate Eigenvector (after 10 iterations with b) ='),
disp(x10)
disp('Approximate Eigenvector (after 100 iterations with b) ='),
disp(x100)
r = rand(size(b)); % Random vector
x10 = evec(A, r, 10);
x100 = evec(A, r, 100);
disp('Approximate Eigenvector (after 10 iterations with random
vector) ='), disp(x10)
disp('Approximate Eigenvector (after 100 iterations with random
vector) ='), disp(x100)
function [x] = evec(A,x,n)
for i = 1:n
x = A*x;
x = x/norm(x);
end
end
Output:
Add Answer to:
Intro Step 11
We will be working with the following matrix [1 2 3 0 0...
Let A be an invertiblen x n matrix and be an eigenvalue of A. Then we...
Let A be an invertiblen x n matrix and be an eigenvalue of A. Then we know the following facts. 1) We have jk is an eigenvalue of A* 2) We have 1 -1 is an eigenvalue of A-1 If 1 = 5 is an eigenvalue of the matrix A, find an eigenvalue of the matrix (A? +41) -'. Enter your answer using three decimal places. Hint: First find an eigenvalue of A² +41. You might do this by assuming...
Problem 1: The matrix e^A is defined by the Mclaurin series e^A=1+A/1!+A^2/2!+A^3/3!+........+A^n/n! This computation ca...
Problem 1:
The matrix e^A is defined by the Mclaurin series
e^A=1+A/1!+A^2/2!+A^3/3!+........+A^n/n!
This computation can be quite difficult, as this problem
illustrates.
a)Enter the function
function E=matexp(A)
E=zeros(size(A));
F=eye(size(A));
k=1;
while norm (E+F-E,1) > 0
E=E+F;
F=A*F/k;
k=k+a;
end
that estimates e^A using Mclaurin series.
b)Apply matexp to the matrix a=[99 -100; 137 -138] to obtaion
matrix A_Mclaurin
c)A_Mclaurin is far from the correct result.ompute the true
value of e^A using the MATLAB function expm as follows:
>>showdemo expmdemo
The...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1....
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1. Give the characteristic polynomial for A in Maple notation in the form t^2 + a*t + b Characteristic polynomial = 2. Find the set of eigenvalues for A, enclosed in braces , ) with the two eigenvalues separated by a comma, like (-4, 7) Set of eigenvalues for A = 5 3. Find one eigenvector for each eigenvalue, using Maple > for vectors, e.g....
Math 3377Quiz 6 Answer the following for the matrix M: 1 2 3 M-0 2 1...
Math 3377Quiz 6 Answer the following for the matrix M: 1 2 3 M-0 2 1 0 0 3 1. What is the characteristic polynomial of M? 2. What are the eigenvalues of M? 3. Show that is an eigenvector of M. What is its eigenvalue? 0
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2. An eigenvector corresponding to the...
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2.
An eigenvector corresponding to the eigenvalue 1 is . Find an
invertible matrix M such that M−1AM= ⎛⎝⎜⎜⎜1000-2000-2⎞⎠⎟⎟⎟. Enter
the Matrix M in the box below.
Question 8: Score 0/2 1 3 -3 4 6 -6 8 The matrix A = 1-6 6 | is diagonalisable with eigenvalues 1,-2 and-2. An eigenvector corresponding to the eigenvalue 1 is -2 2 1 0 0 0 0-2 Find an invertible matrix...
on matlab (1) Matrices are entered row-wise. Row commas. Enter 1 2 3 (2) Element A,...
on matlab
(1) Matrices are entered row-wise. Row commas. Enter 1 2 3 (2) Element A, of matrix A is accesser (3) Correcting an entry is easy to (4) Any submatrix of Ais obtained by d row wise. Rows are separated by semicolons and columns are separated by spaces ner A l 23:45 6. B and hit the return/enter kry matrix A is accessed as A Enter and hit the returnerter key an entry is easy through indesine Enter 19...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3,...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
Find (as a unit vector with negative first term) an eigenvector of the matrix corresponding to...
Find (as a unit vector with negative first term) an eigenvector
of the matrix
corresponding to the eigenvalue lambda = 2
2 – 30 – 6 Find (as a unit vector with negative first term) an eigenvector of the matrix 0 2 0 corresponding to the eigenvalue 1 = 2 0 - 6 4 -4 1/3 x Preview Answer: 6V154 77 V154 154 3V154 154
Review 4: question 1 Let A be an n x n matrix. Which of the below...
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
Let A be an invertiblen x n matrix and be an eigenvalue of A. Then we know the following facts. 1) We have jk is an eigenvalue of A* 2) We have 1 -1 is an eigenvalue of A-1 If 1 = 5 is an eigenvalue of the matrix A, find an eigenvalue of the matrix (A? +41) -'. Enter your answer using three decimal places. Hint: First find an eigenvalue of A² +41. You might do this by assuming...
Problem 1:
The matrix e^A is defined by the Mclaurin series
e^A=1+A/1!+A^2/2!+A^3/3!+........+A^n/n!
This computation can be quite difficult, as this problem
illustrates.
a)Enter the function
function E=matexp(A)
E=zeros(size(A));
F=eye(size(A));
k=1;
while norm (E+F-E,1) > 0
E=E+F;
F=A*F/k;
k=k+a;
end
that estimates e^A using Mclaurin series.
b)Apply matexp to the matrix a=[99 -100; 137 -138] to obtaion
matrix A_Mclaurin
c)A_Mclaurin is far from the correct result.ompute the true
value of e^A using the MATLAB function expm as follows:
>>showdemo expmdemo
The...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1. Give the characteristic polynomial for A in Maple notation in the form t^2 + a*t + b Characteristic polynomial = 2. Find the set of eigenvalues for A, enclosed in braces , ) with the two eigenvalues separated by a comma, like (-4, 7) Set of eigenvalues for A = 5 3. Find one eigenvector for each eigenvalue, using Maple > for vectors, e.g....
Math 3377Quiz 6 Answer the following for the matrix M: 1 2 3 M-0 2 1 0 0 3 1. What is the characteristic polynomial of M? 2. What are the eigenvalues of M? 3. Show that is an eigenvector of M. What is its eigenvalue? 0
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2.
An eigenvector corresponding to the eigenvalue 1 is . Find an
invertible matrix M such that M−1AM= ⎛⎝⎜⎜⎜1000-2000-2⎞⎠⎟⎟⎟. Enter
the Matrix M in the box below.
Question 8: Score 0/2 1 3 -3 4 6 -6 8 The matrix A = 1-6 6 | is diagonalisable with eigenvalues 1,-2 and-2. An eigenvector corresponding to the eigenvalue 1 is -2 2 1 0 0 0 0-2 Find an invertible matrix...
on matlab
(1) Matrices are entered row-wise. Row commas. Enter 1 2 3 (2) Element A, of matrix A is accesser (3) Correcting an entry is easy to (4) Any submatrix of Ais obtained by d row wise. Rows are separated by semicolons and columns are separated by spaces ner A l 23:45 6. B and hit the return/enter kry matrix A is accessed as A Enter and hit the returnerter key an entry is easy through indesine Enter 19...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
Find (as a unit vector with negative first term) an eigenvector
of the matrix
corresponding to the eigenvalue lambda = 2
2 – 30 – 6 Find (as a unit vector with negative first term) an eigenvector of the matrix 0 2 0 corresponding to the eigenvalue 1 = 2 0 - 6 4 -4 1/3 x Preview Answer: 6V154 77 V154 154 3V154 154
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
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