A matrix has eigenvalues λ₁ = 10 and λ₂ = 9. After 50 iterations of the power method, the iterate has not converged to the dominant eigenvector. Which explanation is most accurate?
AThe power method only works for symmetric matrices, so this matrix is likely non-symmetric
BConvergence is governed by |λ₂/λ₁| = 0.9; with the two largest eigenvalues so close, convergence is very slow and 50 iterations may not be enough
CThe iteration converged to the wrong eigenvalue because the starting vector must be chosen carefully
D50 iterations is always sufficient for any matrix of reasonable size
The convergence rate of the power method is |λ₂/λ₁|. When this ratio is close to 1 (here 0.9), the component corresponding to v₂ decays only 10% per iteration, so convergence is very slow. The power method works for non-symmetric matrices and does not require special initialization (a random start is fine); 50 iterations is nowhere near a universal guarantee.
Question 2 Multiple Choice
What is the mathematical role of the normalization step (dividing by the vector's norm) in each power method iteration?
AIt ensures the iteration finds the smallest eigenvalue by projecting out the dominant component
BIt prevents the vector from growing without bound or shrinking to zero, preserving the direction signal while keeping numerical values manageable
CIt accelerates convergence by directly projecting the vector onto the dominant eigenvector subspace
DIt eliminates components corresponding to all non-dominant eigenvectors in a single step
Without normalization, the vector magnitude grows as |λ₁|ᵏ (or shrinks if |λ₁| < 1), causing numerical overflow or underflow. Normalization keeps the magnitude at 1 so the direction converges visibly without arithmetic problems. It does not project out non-dominant components — those decay gradually through many iterations.
Question 3 True / False
The power method converges to the eigenvector corresponding to the eigenvalue with the largest absolute value, not necessarily the algebraically largest eigenvalue.
TTrue
FFalse
Answer: True
Absolute value is what matters. If a matrix has eigenvalues λ₁ = −10 and λ₂ = 5, the power method converges to the eigenvector for λ = −10 because |−10| = 10 > |5| = 5. Students often conflate 'dominant' with 'most positive,' but the method amplifies the direction with the fastest geometric growth, which is determined by magnitude.
Question 4 True / False
If the initial vector v⁰ is chosen to be exactly orthogonal to the dominant eigenvector v₁ (so that the coefficient c₁ = 0 in the eigenvector expansion), the power method will still eventually converge to v₁ due to the normalization step.
TTrue
FFalse
Answer: False
In exact arithmetic, if c₁ = 0, then Aᵏv⁰ has no v₁ component at all, and the iteration converges to whichever eigenvector is dominant in the remaining components. In floating-point arithmetic, rounding errors typically introduce a tiny v₁ component that eventually dominates, but this is a practical accident, not a guarantee. Correct initialization should not rely on this.
Question 5 Short Answer
Why does the power method converge more slowly when the two largest eigenvalues are close in magnitude? Explain using the structure of the iteration.
Think about your answer, then reveal below.
Model answer: Convergence rate is |λ₂/λ₁|. When this ratio is close to 1, the v₂ component of the iterate decays only slightly per step, so many iterations are needed to wash it out. Conversely, when eigenvalues are well-separated (ratio close to 0), the v₂ component decays rapidly and the iterate quickly aligns with v₁.
Decompose v⁰ in the eigenvector basis: after k multiplications, Aᵏv⁰ = λ₁ᵏ(c₁v₁ + c₂(λ₂/λ₁)ᵏv₂ + ...). The second term vanishes at rate |λ₂/λ₁|ᵏ. If λ₁ = 100 and λ₂ = 1, the ratio is 0.01 and convergence is extremely fast. If λ₁ = 10 and λ₂ = 9, the ratio is 0.9 and each step removes only 10% of the contamination from the v₂ direction.