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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: