# What is an eigenvector?

Oct 21, 2015

If vector $v$ and linear transformation of a vector space $A$ are such that $A \left(v\right) = k \cdot v$ (where constant $k$ is called eigenvalue), $v$ is called an eigenvector of linear transformation $A$.

#### Explanation:

Imagine a linear transformation $A$ of stretching all vectors by a factor of $2$ in the three-dimensional space. Any vector $v$ would be transformed into $2 v$. Therefore, for this transformation all vectors are eigenvectors with eigenvalue of $2$.

Consider a rotation of a three-dimensional space around Z-axis by an angle of ${90}^{o}$. Obviously, all vectors except those along the Z-axis will change the direction and, therefore, cannot be eigenvectors. But those vectors along Z-axis (their coordinates are of the form $\left[0 , 0 , z\right]$) will retain their direction and length, therefore they are eigenvectors with eigenvalue of $1$.

Finally, consider a rotation by ${180}^{o}$ in a three-dimensional space around Z-axis. As before, all vectors long Z-axis will not change, so they are eigenvectors with eigenvalue of $1$.
In addition, all vectors in the XY-plane (their coordinates are of the form $\left[x , y , 0\right]$) will change the direction to opposite, while retaining the length. Therefore, they are also eigenvectors with eigenvalues of $- 1$.

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 $A$
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |

Such a matrix, multiplied by any vector $v = \left\{x , y , z\right\}$ will produce $A \cdot v = \left\{2 x , 2 y , 2 z\right\}$
This is obviously equals to $2 \cdot v$. So, we have
$A \cdot v = 2 \cdot v$,
which proves that any vector $v$ is an eigenvector with an eigenvalue $2$.

The second example (rotation by ${90}^{o}$ around Z-axis) can be described as multiplication by a matrix $A$
| 0 | -1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
Such a matrix, multiplied by any vector $v = \left\{x , y , z\right\}$ will produce $A \cdot v = \left\{- y , x , z\right\}$,
which can have the same direction as original vector $v = \left\{x , y , z\right\}$ only if $x = y = 0$, that is if original vector is directed along the Z-axis.