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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

## Eigenspaces 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?

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

### How to prove that a matrix is diagonalizable?

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

### Does a zero vector as eigenvector exists?

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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