# topological proof of the Cayley-Hamilton theorem

We begin by showing that the theorem is true if the characteristic
polynomial^{} does not have repeated roots, and then prove the general case.

Suppose then that the discriminant^{} of the characteristic polynomial is
non-zero, and hence that $T:V\to V$ has $n=dimV$ distinct
eigenvalues^{} once we extend^{1}^{1}Technically, this means that we must
work with the vector space^{} $\overline{V}=V\otimes \overline{k}$, where $\overline{k}$ is the
algebraic closure^{} of the original field of scalars, and with
$\overline{T}:\overline{V}\to \overline{V}$ the extended automorphism with action
$$\overline{T}(v\otimes a)\to T(V)\otimes a,v\in V,a\in \overline{k}.$$
to the algebraic closure of the ground
field.
We can therefore choose a basis of eigenvectors^{}, call them
${\mathbf{v}}_{1},\mathrm{\dots},{\mathbf{v}}_{n}$, with ${\lambda}_{1},\mathrm{\dots},{\lambda}_{n}$ the
corresponding eigenvalues. From the definition of characteristic
polynomial we have that

$${c}_{T}(x)=\prod _{i=1}^{n}(x-{\lambda}_{i}).$$ |

The factors on the right commute, and hence

$${c}_{T}(T){\mathbf{v}}_{i}=0$$ |

for all $i=1,\mathrm{\dots},n$. Since ${c}_{T}(T)$ annihilates a basis, it must, in fact, be zero.

To prove the general case, let $\delta (p)$ denote the discriminant of
a polynomial^{} $p$, and let us remark that the discriminant mapping

$$T\mapsto \delta ({c}_{T}),T\in \mathrm{End}(V)$$ |

is polynomial on
$\mathrm{End}(V)$. Hence the set of $T$ with distinct eigenvalues is a dense
open subset of $\mathrm{End}(V)$ relative to the Zariski
topology^{}. Now the characteristic polynomial map

$$T\mapsto {c}_{T}(T),T\in \mathrm{End}(V)$$ |

is a polynomial map on the vector space $\mathrm{End}(V)$. Since it vanishes on a dense open subset, it must vanish identically. Q.E.D.

Title | topological proof of the Cayley-Hamilton theorem^{} |
---|---|

Canonical name | TopologicalProofOfTheCayleyHamiltonTheorem |

Date of creation | 2013-03-22 12:33:22 |

Last modified on | 2013-03-22 12:33:22 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 7 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 15-00 |