# Example of Eigenvalues and Eigenvectors MATLAB

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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: 