### Eigenvalue and Eigenvector

##### 1. Definition

The linear transformation **T**(`x`_{1}, `x`_{2}) = (5`x`_{1}, 15`x`_{2}) has a simple geometrical meaning: stretch `x`_{1}-direction 5 times and stretch `x`_{2}-direction 15 times. In this exercise, we saw other examples of linear transformations with clear geometrical meanings.

In contrast, the linear transformation **T**(`x`_{1}, `x`_{2}) = (13`x`_{1} - 4`x`_{2}, - 4`x`_{1} + 7`x`_{2}) does not appear to have simple geometrical meaning. However, by introducing a basis **v**_{1} = (1, 2), **v**_{2} = (-2, 1), we get

**T**(**v**_{1}) = (13×1 - 4×2, - 4×1 + 7×2) = (5, 10) = 5**v**_{1},

**T**(**v**_{2}) = (13×(-2) - 4×1, - 4×(-2) + 7×1) = (-30, 15) = 15**v**_{2}.

In other words, **T** stretches **v**_{1}-direction and **v**_{2}-direction by 5 and 15 times, respectively. From the viewpoint of a new basis, **T** is geometrically as simple as the last one.

For another example, the linear transformation **T**: **R**^{3} → **R**^{3} given by the matrix

**A** = [ |
2/3 |
-1/3 |
-1/3 |
] |

-1/3 |
2/3 |
-1/3 |

-1/3 |
-1/3 |
2/3 |

appears to be messy. From this exercise, however, we have **T**(**v**_{1}) = **v**_{1}, **T**(**v**_{2}) = **v**_{2}, **T**(**v**_{3}) = **0** for the basis **v**_{1} = (-1, 1, 0), **v**_{2} = (-1, 0, 1), **v**_{3} = (1, 1, 1). Thus **T** preserves (i.e., multiplying 1) in **v**_{1} and **v**_{2} directions and kills (i.e., multiplying 0) in **v**_{3} direction.

The understanding above is very useful for answering many questions about the linear transformation. For example, the powers of **T**(`x`_{1}, `x`_{2}) = (13`x`_{1} - 4`x`_{2}, - 4`x`_{1} + 7`x`_{2}) is characterized by

**T**^{k}(**v**_{1}) = 5^{k}**v**_{1},
**T**^{k}(**v**_{2}) = 15^{k}**v**_{2},

so that

**T**^{k}(0, 5) = **T**^{k}(2**v**_{1} + **v**_{2}) = 2**T**^{k}(**v**_{1}) + **T**^{k}(**v**_{2}) = 2×5^{k}(1, 2) + 15^{k}(-2, 1) = 5^{k}(2 - 3^{k}2, 4 + 3^{k}).

A direct computation would be

[ |
13 |
-4 |
]^{k} [ |
0 |
], |

-4 |
7 |
5 |

which is impractical to do.

Similarly, direct computation of the powers of the 3 by 3 matrix **A** above is almost impossible. However, the geometric meaning tells us that **A**^{k} = **A** for any `k`.

The key for the good geometrical understanding of a linear transformation is the directions for which the transformation is simply a scalar multiple. This leads to the following definition.

Let **T**: **V** → **V** be a linear transformation. If `λ` is a number and **v** ≠ **0** is a nonzero vector such that

**T**(**v**) = `λ`**v**,

then `λ` is an eigenvalue and **v** is an eigenvector.

Note that **v** ≠ **0** is required because the zero vector does not provide a direction. Moreover, an eigenvector is always associated with an eigenvalue. On the other hand, an engenvalue is always the scaling factor in some eigenvector direction. The two concepts are always bound together.