Subspace- the eigenspace definition

Question

Can the subspace, $null(A^T)$ be referred to as the eigenspace? I was studying the definitions for null space, row space, column space and eigenspace. Since the first three were the first three of the four fundamental subspaces I just assumed $null(A^T)$ was the eigenspace, but looking up eigenspace in my textbook it refers to a section we haven't even covered so I was wanting to get this clarified. If the $null(A^T)$ is not the eigenspace, then what is the definition of the eigenspace which is a subspace?

Answer 1

Eigenspaces are always associated with eigenvalues of a matrix. There is no such thing called "the eigenspace of a matrix" without referring to anything else. Also, while the four fundamental spaces are defined for matrices of any size, eigenvalues (and thus eigenspaces) are defined only for square matrices.

Let $A$ be an $n\times n$ matrix and suppose $\lambda$ is an eigenvalue of $A$. The eigenspace with respect to $\lambda $ is $Null(A-\lambda I)$. In particular, if $\lambda=0$ is an eigenvalue of $A^T$, then $Null(A^T)$ is the eigenspace associate with the eigenvalue $0$.

Answer 2

The subspace null$(A^T)$ is called the left nullspace, since $$ A^Tx = 0 \implies yA = 0,$$ where $y = x^T$. It is indeed the fourth fundamental subspace.

An eigenspace for the eigenvalue $\lambda$ is the nullspace of the matrix $$A - \lambda I,$$ where $I$ is the identity matrix.