Definitions from Eigenvalues and eigenvectors

Eigenvalue

The equation says that the line through $v$ is mapped into itself, with every vector on that line scaled by $\lambda$. A negative real $\lambda$ reverses direction and $\lambda=0$ collapses the line to the origin. For a matrix $A$, the same condition is

Eigenvector

The zero vector is excluded because $T(0)=\lambda 0$ holds for every scalar, so it says nothing about a direction. Any non-zero scalar multiple of an eigenvector is another eigenvector for the same eigenvalue.

Eigenspace

To find the eigenspace, solve the homogeneous system

The kernel description makes the eigenspace a subspace, so it contains the zero vector even though the zero vector is not an eigenvector.

Characteristic polynomial

The characteristic polynomial turns the search for a non-zero solution of $(A-\lambda I)v=0$ into root-finding over the chosen field. A polynomial with no roots in $F$ may gain roots when the scalars are extended, for example from $\mathbb R$ to $\mathbb C$.

Some texts define the polynomial as $\det(\lambda I-A)$ instead. For an $n\times n$ matrix this is $(-1)^n p_A(\lambda)$, so both conventions give the same eigenvalues.

Two names clash with Calculus. The characteristic equation of a scalar ODE is a different object with the same name. In constrained extrema, the letter $\lambda$ denotes a Lagrange multiplier rather than an eigenvalue.