Quiz — Chapter 23: Eigenvalues and Eigenvectors
Twelve quick conceptual checks. Answer before expanding. Each answer includes a one-line why.
Q1. In one sentence, what is an eigenvector of a matrix?
Answer
A nonzero vector whose direction the matrix does not change — the matrix only scales it (by the eigenvalue). *Why: the eigen-equation $A\mathbf{v}=\lambda\mathbf{v}$ says the output is a scaled copy of the input, so $\mathbf{v}$ stays on its own line.*Q2. Why is the zero vector never counted as an eigenvector?
Answer
Because $A\mathbf{0}=\mathbf{0}=\lambda\mathbf{0}$ holds for *every* $\lambda$, so allowing $\mathbf{0}$ would make every scalar an eigenvalue of every matrix — the concept would carry no information. *Why: we require eigenvectors to be nonzero by definition.*Q3. A matrix has an eigenvalue $\lambda = 0$. What does this tell you about the matrix?
Answer
The matrix is **singular** (non-invertible), because the eigenspace $E_0=N(A)$ is nontrivial. *Why: $A\mathbf{v}=0\cdot\mathbf{v}=\mathbf{0}$ for a nonzero $\mathbf{v}$ means $A$ has a nonzero null vector, so $\det(A)=0$.*Q4. If $\mathbf{v}$ is an eigenvector with eigenvalue $5$, is $3\mathbf{v}$ also an eigenvector? With what eigenvalue?
Answer
Yes — $3\mathbf{v}$ is an eigenvector with the **same** eigenvalue $5$. *Why: $A(3\mathbf{v})=3(A\mathbf{v})=3(5\mathbf{v})=5(3\mathbf{v})$; the eigenvalue belongs to the whole line, not to one arrow.*Q5. What does a negative eigenvalue, say $\lambda=-2$, mean geometrically?
Answer
The eigenvector is flipped to point the opposite way *and* scaled by $|\lambda|=2$; it still stays on its own line. *Why: a line through the origin is invariant regardless of which way the arrow on it points.*Q6. Why does a $90°$ rotation matrix have no real eigenvectors?
Answer
Because it turns *every* direction by a quarter-turn, so no real arrow comes back on its own line. *Why: its characteristic equation is $\lambda^2+1=0$, whose roots $\pm i$ are not real — the geometry (no invariant direction) and the algebra (no real root) are the same fact.*Q7. True or false: every $2\times2$ matrix has two distinct real eigenvalues.
Answer
**False.** A matrix may have a repeated eigenvalue (e.g. a shear, $\lambda=1$ twice), complex eigenvalues (e.g. a rotation, $\pm i$), or two distinct real ones. Only the last case gives "two distinct real eigenvalues." *Why: the characteristic polynomial is a quadratic, and quadratics can have a double root or complex roots.*Q8. What is the eigenspace $E_\lambda$, and does it contain the zero vector?
Answer
$E_\lambda=\{\mathbf{v}:A\mathbf{v}=\lambda\mathbf{v}\}=N(A-\lambda I)$, the set of all eigenvectors for $\lambda$ together with $\mathbf{0}$. **Yes**, it contains $\mathbf{0}$ — it is a subspace, and every subspace contains the zero vector. *Why: the eigenvectors are the nonzero members of this subspace; the subspace itself includes $\mathbf{0}$.*Q9. The eigenvalues of a $2\times2$ matrix are $4$ and $-1$. What are its trace and determinant?
Answer
Trace $=4+(-1)=3$; determinant $=4\times(-1)=-4$. *Why: for a $2\times2$ matrix, the eigenvalues sum to the trace and multiply to the determinant.*Q10. When you apply $A$ many times to a vector, which eigen-direction comes to dominate?
Answer
The one with the **largest $|\lambda|$** (the dominant eigenvector). *Why: $A^k\mathbf{v}=\lambda^k\mathbf{v}$, so the direction whose $|\lambda|$ is biggest grows fastest and eventually swamps the rest — the principle behind power iteration and PageRank.*Q11. numpy's np.linalg.eig returns an eigenvector that differs from the one you found by hand. Did one of you make a mistake?
Answer
Not necessarily — eigenvectors are only determined up to a nonzero scalar, and numpy returns *unit-length* eigenvectors (and may flip the sign). If your vector is a scalar multiple of numpy's, you both have the same eigen-direction. *Why: any nonzero multiple of an eigenvector is an eigenvector with the same eigenvalue.*Q12. In one sentence, why do we say eigenvalues "reveal what a matrix really does"?