Finding Eigen Values and Eigen Vectors using MATLAB

Introduction

In this article you will learn about different examples for finding the eigen values and eigen vectors of matrices using different examples on MATLAB.

For the given matrix A and B, find the determinant using these two methods on MATLAB.

  • Using command det(A)
  • By multiplying the diagonal entries.

matrix

A [1 2 3 0; 2 1 4 1; -2 -1 0 1;-1 0 -2 3];

disp(‘the original matrix is A’)

disp (A)

det (A)

ans=-46

Through diagonal entries:

A=[1 2 3 0;2 1 4 1;-2 -1  0 1;-1 0 -2 3]

A(2,:)=A(2,:)-2*A(1,:)

A(3,:)=A(3,:) +2*A(1,:)

A(4,:)=A(4,:)+A(1,:)

A(3,:) = A(3,:) + A(2,:);

A(4,:) = A(4,:) – A(4,2)*A(2,:)/A(2,2)

A(4,:) = A(4,:) -A(4,3)*A(3,:)/A(3,3)

A(1,1)*A(2,2)*A(3,3)*A(4,4)

A =

1     2     3     0

2     1     4     1

-2    -1     0     1

-1     0    -2     3 

A =

1     2     3     0

0    -3    -2     1

-2    -1     0     1

-1     0    -2     3

A =

1     2     3     0

0    -3    -2     1

0     3     6     1

-1     0    -2     3 

A =

1     2     3     0

0    -3    -2     1

0     3     6     1

0     2     1     3

A =

1.0000    2.0000    3.0000         0

0             -3.0000   -2.0000    1.0000

0              0                4.0000    2.0000

0              0               -0.3333    3.6667

A =

 

1.0000    2.0000    3.0000         0

0              -3.0000   -2.0000    1.0000

0               0              4.0000    2.0000

0               0               0    3.8333

ans =

-46

eigen values of matrix using MATLABB=[1 0 2 3; -1 -2 3 2 4 -2 0 3; 1 2 0 -3];

disp(‘the original matríx is B’)

disp (B)

det (B)

ans=-110

Through diagonal entries:

B= [1 0 2 3; -1 -2 3 2; 4 -2 0 3; 1 2 0 -3]

B(2,:)=B(2,:)+B(1,:)

B(3,:)=B(3,:) -4*B(1,:)

B(3,:)=B(3,:)- B(3,2)*B(2,:)/B(2,2)

B(4,:)=B(4,:)-B(1,:)

B(4,:) = B(4,:) – B(4,2)*B(2,:)/B(2,2)

B(4,:) = B(4,:) – B(4,3)*B(3,:)/B(3,3)

B(1,1)*B(2,2)*B(3,3)*B(4,4)

B =

1     0     2     3

-1    -2     3     2

4    -2     0     3

1     2     0    -3

B =

1     0     2     3

0    -2     5     5

4    -2     0     3

1     2     0    -3

B =

1     0     2     3

0    -2     5     5

0    -2    -8    -9

1     2     0    -3

B =

1     0     2     3

0    -2     5     5

0     0   -13   -14

1     2     0    -3

B =

1     0     2     3

0    -2     5     5

0     0   -13   -14

0     2    -2    -6

B =

1     0     2     3

0    -2     5     5

0     0   -13   -14

0     0     3    -1

B =

1.0000         0    2.0000    3.0000

0   -2.0000    5.0000    5.0000

0         0  -13.0000  -14.0000

0         0         0   -4.2308

ans =

-110

For the matrices A and B, find its characteristic polynomial, Eigen values and trace of A and B using MATLAB.

matrix A and B eigen values and eigen vectors using MARLAB

A

A=[-9 -2 -10; 3 2 3; 8 2 9];

disp(“matrix A is”)

disp(A)

disp(“characteristic polynomial of A is”)

poly(A)

disp(“Eigen values of A is”)

eig(A)

disp(“trace of B is”)

trace(A)

Answer:

Characteristic polynomial of B is

ans =

1.0000   -2.0000   -1.0000    2.0000

eigen values of A is

ans =

-1.0000

2.0000

1.0000

trace of B is

ans =

2

B

B=[2 1 -5 2; 1 2 13 2;0 0 3 -1; 0 0 1 1];

disp(“matrix B is”)

disp(B)

disp(“characteristic polynomial of B is”)

poly(B)

disp(“Eigen values of B is”)

eig(B)

disp(“trace of B is”)

trace(B)

Answer:

characteristic polynomial of B is

ans =

1    -8    23   -28    12

eigen values of B is

ans =

3.0000

1.0000

2.0000

2.0000

trace of B is

ans =

8

What do the following commands do?

Explain the output of each

  • e = eig(A)

This function gives the eigon values

  • [V,D] = eig(A)

V =

0.7845    0.6667   -0.7071

-0.1961   -0.3333   -0.0000

-0.5883   -0.6667    0.7071

D =

-1.0000         0         0

0    2.0000         0

0         0    1.0000

  • [V,D,W] = eig(A)

V =

0.7071   -0.7071   -0.9764   -0.9764

0.7071    0.7071    0.1953    0.1953

0         0    0.0651    0.0651

0         0    0.0651    0.0651

D =

3.0000         0         0         0

0    1.0000         0         0

0         0    2.0000         0

0         0         0    2.0000

W =

0.0463   -0.0554         0         0

0.0463    0.0554         0         0

0.9265    0.0000    0.7071   -0.7071

-0.3706   -0.9969   -0.7071    0.7071

Also read here

 

what are the eigen values and eigen vectors? explain with examples

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