Are vectors in the null space of a matrix considered eigenvectors?

Question

From what I've learned about the definition of an eigenvector, it seems like a vector that gets mapped to zero should just be considered an eigenvector where $\lambda = 0$. Is that true, or are those considered a special case?

Answer 1

With the exception of the zero vector any vector $v\in\operatorname{null}T$ are eigenvectors of $T$.

Answer 2

Yes, this is true. Indeed, an eigenvector of $A$ with eigenvalue $\lambda$ is in the null space of $A-\lambda I$.

Answer 3

Yes it is correct by the definition, for $\vec x\neq 0$

$$A\vec x=0\vec x$$

then $\vec x$ is an eigenvector with eigenvalue $\lambda=0$.

Answer 4

Yes, they are eigenvectors (unless it's the null vector).

Answer 5

Just adding some more perspective -

it is not a special case, Definition itself clarify -

Any non-zero vector that remains in its own span under a transformation is Eigen vector, i.e. any nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it

Note that since Zero-vector can be expressed as Linear Combination of any other vector hence its the only vector that can be considered as parallel to any other vector.

therefore, when Transformation matrix A is Singular, that means its Null-space is more than Trivial, and Determinant of A is 0, then we will surely get at least one Eigen-value as 0, and eigen vector correspond to eigen-value 0 is nothing but Null-space of A { except trivial vector }, because any vector in Null-space of A will land on Zero-vector (origin) post transformation . hence technically those vectors will remain in their own span.