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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 -

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# Eigenvectors of a Matrix

## Eigenvectors Calculator

### What are eigen vectors of a matrix? (Definition)

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.

### How to calculate eigenvectors 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 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 $\vec{v_2} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$.

### Why use the eigenvectors of a matrix?

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

### 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.

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).

### Does a zero vector as an eigenvector exist?

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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