Extension of the field of a matrix vector space

Question

A pretty common step that I'm encountering in linear algebra proofs is "extending the field" in which a matrix is defined. I'll make some examples:

Cayley-Hamilton theorem

The book I'm using proves Cayley-Hamilton theorem for triangulable endomorphism, and than it states that it's valid for any endomorphism because a matrix over a field $F$ can be seen as a matrix over the algebraic closure of that field $F'$. Clearly the algebraic closure is algebrically closed(and the water is wet), so any matrix is triangulable over $F'$ and clearly the characteristic polynomial doesn't depend upon the chosen field($F$ or $F'$). So any matrix over any field is a zero of its characteristic polynomial and by isomorphism every endomorphism is a zero of its characteristic polynomial.

Existence of eigenvalues of a symmetric endomorphism

My book proves first that an hermitian endomorphisms has real eigenvalues. Then it states that a real symmetric matrix can be seen as an hermitian matrix over $\mathbb{C}$, so it has only real eigenvalues(that are also eigenvalues of the starting matrix over real numbers, since they are real). So any symmetric matrix has eigenvalues and by isomorphism every symmetric endomorphism has eigenvalues.

In general I don't like proofs with matrices(I know, I'm quirky), I prefer working always with endomorphisms. Clearly I perfectly know that matrix and endomorphisms algebras are isomorphic and it's because of this, that I think that we should be able to complete the proofs above without the use of matrices, but only with endomorphisms. Is there a way to "extend the field of scalars" to make this kind of passage meaningful also in "endomorphism language".

Thank you in advance.

Answer 1

The answer is yes. In the most general setting, extending the field of scalars is done via the (appropriately named) extension of scalars. Of particular note is the notion of complexification. Note also that a "complexified" real vector space is sometimes referred to as a linear complex structure.

Whatever your field $F$, it is not too difficult to establish that there exists a sensible notion of multiplication by the extended field $F'$. The "tricky" parts are usually related to how the original vector space "fits" within this extended structure. For example: the complex eigenvalues and eigenvectors of a real matrix come in complex-conjugate pairs; coming up with (and proving) an analogous statement for endomorphisms over a complexified vector space means that we need to say what exactly "having real entries" should mean in the abstracted context.