In the equation AX = λX, what is X called?

• A. Basis vector.
• B. Column vector.
• C. Null vector.
• D. Eigenvector.

Answer: (D)

When a matrix A acts on a vector X and the result is a scalar multiple of X, X is an eigenvector corresponding to the eigenvalue λ. In AX = λX, the transformation by A changes the magnitude of X by λ but keeps its direction—the vector is stretched or compressed along the same line. This nonzero vector X that only scales under the transformation is the eigenvector. The associated scalar λ is the eigenvalue. Remember, the zero vector would satisfy AX = λX for any λ, but eigenvectors are defined to be nonzero. In practice, you find eigenvectors and eigenvalues by solving (A − λI)X = 0 with X ≠ 0.