Jordan normal form

This is a square matrix.

A is the algebraically closed field. In mathematics, a field F is algebraically closed if every non-constant polynomial in F[x] has a root in F. Moreover, P is the regular matrix. J is called Jordan normal form.

Ae1=5e1

Ae2=e1+5e2

Ae3=e2+5e3

You can write this.

A5e1=0

A5e2=e1

A5e3=e2

∴
e3→e2→e1→0

e is an eigenvector of A.

This is called Computational Topology.