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Task:

Find the eigenvectors and eigenvalues of the following matrix in MATLAB:

     

Solution:

MATLAB can compute eigenvalues and eigenvectors of a square matrix, either numerically or symbolically.

Numerical eigenvalues and eigenvectors

(Note: green color marks user input, and blue color is MATLAB response.)

First let’s set matrix:

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

 

A =

 

The “eig” command computes the eigenvalues and eigenvectors:

 

[V,D] = eig(A)

 

V =
-0.49410-0.558050.66667
-0.472020.816140.33333
0.730110.149980.66667
D =
Diagonal Matrix
-1.0000000
0-1.000000
008.00000

 

The “eig” command returns two matrices. The first contains the eigenvectors as the columns of the matrix, while the second is a diagonal matrix with the eigenvalues on the diagonal. The eigenvectors and eigenvalues are given in the same order.

 

We can also call the “eig” command with a single output, in which case only the eigenvalues are returned, and in a vector instead of a matrix:

ev = eig(A)

ev =

-1.00000

-1.00000

8.00000

Symbolical eigenvalues and eigenvectors

To obtain symbolic (exact) eigenvalues and eigenvectors, it is only necessary to define the matrix to be symbolic:

The computation then proceeds exactly as before: