# What is an eigenvector?

##### 1 Answer

#### Answer:

If vector *eigenvalue*), *eigenvector* of linear transformation

#### Explanation:

Imagine a linear transformation *eigenvectors* with *eigenvalue* of

Consider a rotation of a three-dimensional space around Z-axis by an angle of *eigenvectors*. But those vectors along Z-axis (their coordinates are of the form *eigenvectors* with *eigenvalue* of

Finally, consider a rotation by *eigenvectors* with *eigenvalue* of

In addition, all vectors in the XY-plane (their coordinates are of the form *eigenvectors* with *eigenvalues* of

Any linear transformation of a vector space can be expressed as multiplication of a vector by a matrix. For instance, the first example of stretching is described as multiplication by a matrix

| 2 | 0 | 0 |

| 0 | 2 | 0 |

| 0 | 0 | 2 |

Such a matrix, multiplied by any vector

This is obviously equals to

which proves that any vector *eigenvector* with an *eigenvalue*

The second example (rotation by

| 0 | -1 | 0 |

| 1 | 0 | 0 |

| 0 | 0 | 1 |

Such a matrix, multiplied by any vector

which can have the same direction as original vector