Tool to calculate eigenvectors of a matrix. Eigenvectors of a matrix are vectors whose direction remains unchanged after multiplying by the matrix. They are associated with an eigenvalue.

Eigenvectors of a Matrix - dCode

Tag(s) : Matrix

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To find **eigenvectors**, take $ M $ a square matrix of size $ n $ and $ \lambda_i $ its eigenvalues. **Eigenvectors** are the solution of the system $ ( M − \lambda I_n ) \vec{X} = \vec{0} $ with $ I_n $ the identity matrix.

__Example:__ The 2x2 matrix $$ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $$

Eigenvalues for the matrix $ M $ are $ \lambda_1 = 5 $ and $ \lambda_2 = -1 $ (see tool for calculating matrices eigenvalues).

For each eigenvalue, look for the associated **eigenvector**.

__Example:__ For $ \lambda_1 = 5 $, solve $ ( M − 5 I_n ) X = \vec{0} $: $$ \begin{bmatrix} 1-5 & 2 \\ 4 & 3-5 \end{bmatrix} . \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ and find as solution $$ \begin{align} -4 x_1 + 2 x_2 &= 0 \\ 4 x_1 - 2 x_2 &= 0 \end{align} \iff \begin{matrix} x_1 = 1 \\ x_2 = 2 \end{matrix} $$ So the **eigenvector** associated to $ \lambda_1 = 5 $ is $ \begin{pmatrix} 1 \\ 2 \end{pmatrix} $

__Example:__ For $ \lambda_2 = -1 $, solve $ ( M + I_n ) X = \vec{0} $ like this: $$ \begin{bmatrix} 1+1 & 2 \\ 4 & 3+1 \end{bmatrix} . \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\ \begin{align} 2 x_1 + 2 x_2 &= 0 \\ 4 x_1 + 4 x_2 &= 0 \end{align} \iff \begin{matrix} x_1 = -1 \\ x_2 = 1 \end{matrix} $$

So the **eigenvector** associated to $ \lambda_1 = -1 $ is $ \begin{pmatrix} -1 \\ 1 \end{pmatrix} $.

A matrix $ M $ matrix of order $ n $ is a diagonalizable matrix if it has $ n $ **eigenvectors** associated with $ n $ distinct eigenvalues.

The definition of the **eigenvector** precludes its nullity. However, if in a calculation the number of independent **eigenvectors** is less than the number of eigenvalues, dCode will sometimes display a null vector.

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Source : https://www.dcode.fr/matrix-eigenvectors

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