I was having a look at a post on algebraic multiplicity (Understanding algebraic Multiplicity in two ways) which mentions that given a complex vector space and any eigenvalue $\lambda$, the dimension of the generalized eigenspace of $\lambda$ is the same as the multiplicity of $\lambda$ as a root of the characteristic polynomial.
Since it is mentioned that the vector space is complex, I am curious to know if I had a real vector space and a real eigenvalue $\lambda$ of a linear operator $T$, would the dimension of the generalized eigenspace of $\lambda$ be necessarily equal to its multiplicity as a root of the characteristic polynomial of $T$?