Questions: Eigenvalues and Eigenvectors

Question 1

Matrix A has eigenvector v with eigenvalue λ = -2. What happens to v when multiplied by A?

• A. v is rotated 180° with no scaling
• B. v is reversed in direction and doubled in length
• C. v is unchanged
• D. v is projected onto a coordinate axis

Answer: (B)

Av = -2v means v is scaled by -2: the negative sign flips the direction (180° reversal) and the magnitude 2 doubles the length. The key insight is that eigenvectors are only scaled — never rotated off their line — and the eigenvalue encodes both direction (sign) and magnitude of that scaling.

Question 2

If v is an eigenvector of A with eigenvalue λ, then the vector 5v is also an eigenvector of A with the same eigenvalue λ.

• T. True
• F. False

Answer: True

A(5v) = 5(Av) = 5(λv) = λ(5v), so 5v satisfies the eigenvector equation with the same λ. Eigenvectors are not unique — any nonzero scalar multiple is also an eigenvector. The eigenspace E_λ = nul(A − λI) captures all of them as a subspace.

Question 3

What is the geometric meaning of an eigenvector of a matrix transformation?

Model answer: An eigenvector lies on a line through the origin that the transformation maps to itself — the vector is only stretched or flipped, never rotated off that line.

Most vectors get both scaled and rotated by a matrix. Eigenvectors are the special directions that remain on the same line after the transformation — only their length (and possibly direction sign) changes. This makes them the 'skeleton' that reveals the matrix's fundamental behavior.