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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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.

Example: Consider the 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.

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 \( \begin{pmatrix} 1 \\ 2 \end{pmatrix} \).

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 \( \begin{pmatrix} -1 \\ 1 \end{pmatrix} \).

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 display a null vector.

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