Why are algebraic and geometric multiplicities called like that?

Question

I'm aware of the formal definition of algebraic and geometric multiplicities, but I can't make much sense out of the names.

What I mean is, if it were me, I would name these two quantities multiplicity one and multiplicity two, for all I know. But I'm guessing that the people who named these two quantities had something intuitive in mind when they defined them as algebraic and geometric. Is it just because the first is referring to a polynomial (which is kind of algebraic) and the second to a dimension (which is kind of geometric), or is it something more profound that I'm missing?

Answer 1

The algebraic multiplicity is referring to multiplicity of some root of the characteristic polynomial, which is defined for matrices over any commutative ring. You can define this independent of a space on which the matrix is acting.

In contrast the geometric multiplicity refers to the dimension of the eigenspace for your eigenvalue, so it is implied that your matrix is acting on a vector space, which is a geometric notion. I'm not sure this is more profound but at least it's an ellaboration on what you were thinking.