# proof of necessary and sufficient condition for diagonalizability

First, suppose that $T$ is diagonalizable^{}. Then $V$ has a basis whose elements $\{{v}_{1},\mathrm{\dots},{v}_{n}\}$ are eigenvectors^{} of $T$ associated with the eigenvalues^{} $\{{\lambda}_{1},\mathrm{\dots},{\lambda}_{n}\}$ respectively. For each $i=1,\mathrm{\dots},n$, as ${v}_{i}$ is an eigenvector, its annihilator polynomial is ${m}_{{v}_{i}}=X-{\lambda}_{i}$. As these vectors form a basis of $V$, we have that the minimal polynomial^{} (http://planetmath.org/MinimalPolynomialEndomorphism) of $T$ is ${m}_{T}=\mathrm{lcm}(X-{\lambda}_{1},\mathrm{\dots},X-{\lambda}_{n})$ which is trivially a product of linear factors.

Now, suppose that ${m}_{T}=(X-{\lambda}_{1})\mathrm{\dots}(X-{\lambda}_{p})$ for some $p$.
Let $v\in V$. Consider the $T$ - cyclic subspace generated by $v$, $$. Let ${T}_{v}$ be the restriction of $T$ to $Z(v,T)$. Of course, $v$ is a cyclic vector of $Z(v,{T}_{v})$, and then ${m}_{v}={m}_{{T}_{v}}={\chi}_{T}$. This is really easy to see: the dimension^{} of $Z(v,T)$ is $r+1$, and it’s also the degree of ${m}_{v}$. But as ${m}_{v}$ divides ${m}_{{T}_{v}}$ (because ${m}_{{T}_{v}}v=0$), and ${m}_{T}$ divides ${\chi}_{{T}_{v}}$ (Cayley-Hamilton theorem^{}), we have that ${m}_{v}$ divides ${\chi}_{{T}_{v}}$. As these are two monic polynomials of degree $r+1$ and one divides the other, they are equal. And then by the same reasoning ${m}_{v}={m}_{{T}_{v}}={\chi}_{T}$.
But as ${m}_{v}$ divides ${m}_{T}$, then as ${m}_{v}={m}_{{T}_{v}}$, we have that ${m}_{{T}_{v}}$ divides ${m}_{T}$, and then ${m}_{{T}_{v}}$ has no multiple roots and they all lie in $k$. But then so does ${\chi}_{{T}_{v}}$. Suppose that these roots are ${\lambda}_{1},\mathrm{\dots},{\lambda}_{r+1}$. Then $Z(v,T)={\oplus}_{{\lambda}_{i}}{E}_{{\lambda}_{i}}$, where ${E}_{{\lambda}_{i}}$ is the eigenspace^{} associated to ${\lambda}_{i}$. Then $v$ is a sum of eigenvectors. QED.

Title | proof of necessary and sufficient condition for diagonalizability |
---|---|

Canonical name | ProofOfNecessaryAndSufficientConditionForDiagonalizability |

Date of creation | 2013-03-22 14:15:45 |

Last modified on | 2013-03-22 14:15:45 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 13 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 15A04 |