The space where the kernel of a real square matrix is defined

Question

Let $A\in\mathbb{R}^{n\times n}$ be a real matrix. From the characteristic equation $Av=\lambda v$, we know that eigenvectors $v$ can be complex, therefore they belong to $\mathbb{C}^n$. Since the null space also comes from the characteristic equation with zero eigenvalue, it seems that the null space should also be defined over the complex field $\mathbb{C}^n$. However, if we define our linear transformation $A:\mathbb{R}^n\to\mathbb{R}^n$, our null space should be a subspace of $\mathbb{R}^n$ (not $\mathbb{C}^n$). Where does this inconsistency come from? I refer to page 6 of Horn's Matrix Analysis for the definition of null space. Any comment or response is greatly appreciated.

Answer 1

If that's the case, we don't call such set a null space of $\mathcal A$. $A \in \mathbb R^{n\times n} \subset \mathbb C^{n\times n}$, so $A$ could be viewed as a real matrix and as a complex matrix simultaneously. But for a linear transformation, we only allow the eigenvectors and the eigenvalues be real, so for example if $\mathcal A \colon \mathbb R^2 \to \mathbb R^2$ is given by the matrix $$ \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, $$ then $\mathcal A$ has no eigenvectors.

For general $\mathcal A \colon V \to V$ where $V$ is a real vector space, the null space is just $\{v \in V \colon \mathcal Av = 0\}$, which always exists.