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Eigenvectors of a Matrix

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.

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Eigenvectors of a Matrix -

Tag(s) : Mathematics,Algebra,Symbolic Computation

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# Eigenvectors of a Matrix

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## Eigenvectors Calculator

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.

### How to calculate eigen vectors of a matrix?

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.

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

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 : $$\begin{pmatrix} 1-5 & 2 \\ 4 & 3-5 \end{pmatrix} . \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

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 $$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$$.