### Polynomials of Matrix

##### 2. Generalized eigenspace

Let `λ`_{1}, `λ`_{2}, ..., `λ`_{k} be all the distinct eigenvalues of an `n` by `n` matrix **A**, with algebraic multiplicities `m`_{1}, `m`_{2}, ..., `m`_{k}. Then by this result, the characteristic polynomial is

`p`(`t`) = `det`(**A** - `λ`**I**) = (`λ`_{1} - `λ`)^{m1} (`λ`_{2} - `λ`)^{m2} ... (`λ`_{k} - `λ`)^{mk}^{}.

By Cayley-Hamilton theorem, we have

`p`(**A**) = (`λ`_{1}**I** - **A**)^{m1} (`λ`_{2}**I** - **A**)^{m2} ... (`λ`_{k}**I** - **A**)^{mk}^{} = **O**.

In particular, we have

In an earlier example, we considered the following diagonalizable matrices

**A** = [ |
13 |
-4 |
], with eigenvalues 5, 15; |

-4 |
7 |

**B** = [ |
1 |
3 |
-3 |
], with eigenvalues 4, -2. |

-3 |
7 |
-3 |

-6 |
6 |
-2 |

For the polynomial `p`(`t`) = (5 - `t`)(15 - `t`) = 75 - 20`t` + `t`^{2}, applying the matrix `p`(**A**) = 75**I** - 20**A** + **A**^{2} to the basis of eigenvectors of **A** is the same as multiplying `p`(`λ`) = `p`(5 or 15) = 0 to the eigenvectors. In other words, applying `p`(**A**) to the basis vectors always give us