We have already seen the use of eigenvalue and eigenvector in the computation of powers of matrix such as

[ | 13 | -4 | ]^{k}. |

-4 | 7 |

In a general *linear* discrete dynamical system, we are given an initial vector **x**_{0} ∈ ` V` and a linear transformation

**x**_{0},

**x**_{1} = ` T`(

:

:

Suppose ` T` is

**x**_{0} = `c`_{1}**v**_{1} + `c`_{2}**v**_{2} + ... + `c _{n}v_{n}`,

and obtain the formula for ` x_{k}`:

` x_{k}` =

Example Suppose the three sequences

`x`_{1}, `x`_{2}, ..., `x _{k}`, ...

start with `x`_{1} = `y`_{1} = `z`_{1} = 1 and are related by the recursive relation

x_{k+1} |
= | x_{k} |
+ 3y_{k} |
- 3z_{k} |

y_{k+1} |
= | - 3x_{k} | + 7y_{k} | - 3z_{k} |

z_{k+1} |
= | - 6x_{k} | + 6y_{k} | - 2z_{k} |

Then we denote

= [x_{k} |
x_{k} |
], = [A |
1 | 3 | -3 | ] |

y_{k} |
-3 | 7 | -3 | |||

z_{k} |
-6 | 6 | -2 |

and get ` x_{k}` =

x_{1} = [ |
1 | ] = (1/2)[ | 1 | ] + 0[ | 1 | ] + (1/2)[ | 1 | ], |

1 | 1 | 0 | 1 | |||||

1 | 0 | -1 | 2 |

we conclude that

= x_{k}A^{k-1}[ |
1 | ] = (1/2)4^{k-1}[ |
1 | ] + (1/2)(-2)^{k-1}[ |
1 | ] = [ | 2×4^{k-2} + (-1)^{k-1}2^{k-2} |
]. |

1 | 1 | 1 | 2×4^{k-2} + (-1)^{k-1}2^{k-2} |
|||||

1 | 0 | 2 | 2×4^{k-2} + (-1)^{k-1}2^{k-1} |

In other words, the three sequences are given by

`x _{k}` = 2×4

Some problems may be converted to discrete dynamical systems.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

is determined by the initial data `a`_{0}
= `a`_{1} = 1 and the recursive relation
`a`_{k+2} = `a`_{k+1}
+ `a _{k}`. We would like to find the general formula for

Denote

= [x_{k} |
a_{k+1} |
]. |

a_{k} |

Then we have

x_{0} = [ |
1 | ], x_{k+1} = [ |
a_{k+2} |
] = [ | a_{k+1} + a_{k} |
] = [ | 1 | 1 | ][ | a_{k+1} |
] = .Ax_{k} |

1 | a_{k+1} |
a_{k+1} |
1 | 0 | a_{k} |

The characteristic polynomial of the matrix is `det`(` A` -

λ_{1} = |
1 + √ | , λ_{2} = |
1 - √ | . |

2 | 2 |

We also find the corresponding eigenvectors

v_{1} = ( |
1 + √ | , 1), v_{2} = ( |
1 - √ | , 1). |

2 | 2 |

From

x_{0} = (1, 1) = |
(1 + √v_{1} - (1 - √ )v_{2} |
), |

2√ |

we then get

= x_{k}A^{k}x_{0} = |
(1 + √λ_{1}^{k}v_{1} - (1 - √ )λ_{2}^{k}v_{2} |
). |

2√ |

Since `a _{k}` is the second coordinate of

a = _{k} |
(1 + √λ_{1} - (1 - √ )^{k}λ_{2}^{k} |
)= | 1 | [ ( | 1 + √ | )^{k+1} - ( |
1 - √ | )^{k+1} |
]. |

2√ | √ | 2 | 2 |