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) : Mathematics,Algebra,Symbolic Computation

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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.

Consider \( M \) 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.

$$ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $$

Eigenvalues for the matrix M are \( \lambda_1 = 5 \) and \( \lambda_2 = -1 \).

You can seek for example the eigenvector associated to \( \lambda_1 = 5 \). You solve \( ( M − 5 I_n ) X = \vec{0} \) so : $$ \begin{bmatrix} 1-5 & 2 \\ 4 & 3-5 \end{bmatrix} . \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$

and you find

$$ \begin{matrix} -4 x_1 + 2 x_2 = 0 \\ 4 x_1 - 2 x_2 = 0 \end{matrix} \iff \begin{matrix} x_1 = 1 \\ x_2 = 2 \end{matrix} $$

The eigenvector associated to \( \lambda_1 = 5 \) is \( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \).

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