Discover Linear Algebra: A first course in linear algebra

Eigenvectors and eigenvalues go together in pairs, with the connection between the two provided by the equality $A\mathbf{x}=\lambda \mathbf{x}\text{.}$ In this situation, we say that the two objects correspond to each other. So we might say that $\mathbf{x}$ is an eigenvector of $A$ that corresponds to the eigenvalue $\lambda \text{.}$ Equivalently, we might say that $\lambda$ is an eigenvalue of $A$ that corresponds to the eigenvector $\mathbf{x}\text{.}$ However, note that an eigenvalue can correspond to many eigenvectors (in fact, an infinite number of them), an eigenvector must correspond to exactly one eigenvalue.

In other resources you may seem the terms characteristic vector, characteristic value, and characteristic space used in place of the terminology introduced above.