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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An eigenvector of a matrix is a characteristic vector (or privileged axis or direction) on which a linear transformation behaves like a scalar multiplication by a constant named eigenvalue.

In other words, these are the vectors that only change by one scale when multiplied by the matrix.

The set of eigenvectors form an eigenspace.

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 matrix eigenvalues).

For each eigenvalue, look for the associated eigenvector.

__Example:__ For $ \lambda_1 = 5 $, solve $ ( M − 5 I_n ) X = \vec{0} $: $$ \begin{bmatrix} 1-5 & 2 \\ 4 & 3-5 \end{bmatrix} . \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ be the equivalent system of equations $$ \begin{align} -4 x_1 + 2 x_2 &= 0 \\ 4 x_1 - 2 x_2 &= 0 \end{align} $$ which admits several solutions including $$ \begin{array}{c} x_1 = 1 \\ x_2 = 2 \end{array} $$ So the eigenvector associated to $ \lambda_1 = 5 $ is $ \vec{v_1} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} $

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

So the eigenvector associated to $ \lambda_2 = -1 $ is $ \vec{v_2} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} $.

Eigenvectors can be used to simplify certain calculations, to understand the linear transformations induced by the matrix and to solve problems related to eigenvalues.

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

That is, it has enough linearly independent eigenvectors to form a basis for the vector space in which it operates (necessary condition for its diagonalization).

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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Cite as source (bibliography):

*Eigenvectors of a Matrix* on dCode.fr [online website], retrieved on 2024-11-11,

eigenvector,matrix,eigenvalue,space,direction,diagonalization,transformation

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

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