A proof of the Cayley-Hamilton theorem using algebraic geometry
DOI:
https://doi.org/10.17268/sel.mat.2021.02.09Keywords:
Affine space, algebraic set, algebraic manifold, Zariski topologyAbstract
In this work, we will prove the Cayley-Hamilton theorem using algebraic geometry. We will see a different proof than the one seen in a linear algebra course, in this case we will use the Zariski topology, then we will take advantage of the fact that every square matrix of order n _ n, with entries in a field K, denoted by (aij)n_n can be seen as an element of the affine space of dimension n _ n over the field K and thanks to this, we can resort to algebraic sets and algebraic varieties in order to obtain some results seen in an algebraic geometry and to get a proof of the Cayley-Hamilton theorem.
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