Please Just answer the questions. One method for finding the eigenvalues and eig
ID: 2963981 • Letter: P
Question
Please Just answer the questions.
One method for finding the eigenvalues and eigenvectors of a matrix is to use the power method. Open Mathlab in my.math.wsu (or your preferred software). Remember to pay close attention to the syntax: spacing, capitol vs. lower case letters, type of bracket ,etc. matter for the code to work. Type in the matrix:
A=[ 6 -4 -8
-3 10 11
2 -4 -4]
The eigenvector associated with the largest eigenvalue can be found by looking at powers of this matrix. For example type in
A^10
to find the 10th power of A. Notice that this matrix has really big numbers in it. One strategy to combat this is to divide by an appropriate number. There are several strategies for choosing this number, however in this case, dividing by 6 works well. Type in
(A/6)^10
Explore what happens when A is powered up even higher by increasing the exponent. (For example, the 20th, 30th, and 50th powers).
What do you notice when you compare different high powers of A/6?
What is the relationship between the columns of these high powers of A/6?
Create a vector x that is equal to the first column of some high power of A/6. For example type:
B=(A/6)^100
x=B(1:3,1)
Now test to see if this vector is an eigenvector for A by typing
A*x
Is x an eigenvector for A? What is the associated eigenvalue?
Explanation / Answer
MAT LAB CODE
clc
clear all
A=[6 -4 -8;-3 10 11;2 -4 -4];
B=A/6;
C=B^(100)
x=C(1:3, 1);
proof=A*x
eigen_values=eig(A)
Results
A/6 =
1.0000 -0.6667 -1.3333
-0.5000 1.6667 1.8333
0.3333 -0.6667 -0.6667
(A/6)^10
0.5173 -1.0000 -1.5173
-0.9913 2.0000 2.9913
0.5000 -1.0000 -1.5000
(A/6)^20
0.5003 -1.0000 -1.5003
-0.9998 2.0000 2.9998
0.5000 -1.0000 -1.5000
(A/6)^30
0.5000 -1.0000 -1.5000
-1.0000 2.0000 3.0000
0.5000 -1.0000 -1.5000
(A/6)^50=
0.5000 -1.0000 -1.5000
-1.0000 2.0000 3.0000
0.5000 -1.0000 -1.5000
x =
0.5000
-1.0000
0.5000
proof =6x (6 is an eigen value)
3.0000
-6.0000
3.0000