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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 =
| 3 | 2 | 4 |
| 2 | 0 | 2 |
| 4 | 2 | 3 |
The “eig” command computes the eigenvalues and eigenvectors:
[V,D] = eig(A)
| V = | ||
| -0.49410 | -0.55805 | 0.66667 |
| -0.47202 | 0.81614 | 0.33333 |
| 0.73011 | 0.14998 | 0.66667 |
| D = | ||
| Diagonal Matrix | ||
| -1.00000 | 0 | 0 |
| 0 | -1.00000 | 0 |
| 0 | 0 | 8.00000 |
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:
