Question bank — Eigenvalue computation — power method, inverse iteration

False — it converges to the largest in magnitude $|\lambda |$. A matrix with eigenvalues $-9$ and $+4$ has dominant $\lambda =-9$, not $4$.

False — the ratio $|{\lambda }_2/{\lambda }_1|=1$ never decays, so those two components persist and the iterate oscillates or rotates within the $v_1,v_2$ plane instead of settling.

False — it only needs a dominant eigenvalue and a full eigenbasis. Symmetry is a bonus: it makes the Rayleigh quotient accurate to $O({\text{error}}^2)$ rather than $O(\text{error})$.

False — power method's rate is $|{\lambda }_2/{\lambda }_1|$ while inverse iteration's is $|{\lambda }_n/{\lambda }_{n-1}|$; these depend on different eigenvalue gaps and are usually very different.

False and dangerous — then $A-\sigma I$ is exactly singular, so the system $(A-\sigma I)y=x$ has no unique solution. In practice near-singularity is fine (even helpful); exact singularity breaks the solve.

False — normalization only rescales length; direction is untouched. It exists purely to stop the vector from overflowing (${\lambda }_1^k$) or underflowing.

False — it is a best-fit eigenvalue for the current vector, so it gives a usable (improving) estimate at every step, well before convergence.

True — from $Av=\lambda v$, multiply by $A^{-1}/\lambda$ to get $A^{-1}v=\frac{1}{\lambda }v$; same $v$, reciprocal eigenvalue. See Spectral Decomposition.

False — $(A-\sigma I)v=(\lambda -\sigma )v$ keeps the same $v$; only eigenvalues slide by $\sigma$. That invariance is exactly why shift-and-invert can target a chosen eigenvector.

The result is correct but the method is wrong: explicitly inverting is $O(n^3)$, numerically unstable, and wasteful. Instead solve $A{y}_{k+1}=x_k$ each step, reusing one LU Decomposition — "solve, don't invert."

$A^k$ growing fast is not convergence of direction. The error in direction decays like $|{\lambda }_2/{\lambda }_1{|}^k$ — a constant ratio per step, i.e. linear convergence, not quadratic.

$c_1$ is the $v_1$-component of the specific seed; a different seed lying in $\text{span}(v_2,\dots ,v_n)$ has $c_1=0$. You must check each seed (random seeds work almost surely).

The rate is the ratio $\left|\frac{{\lambda }_j-\sigma }{{\lambda }_m-\sigma }\right|$ where ${\lambda }_m$ is the second-closest eigenvalue to $\sigma$. Only the ratio (numerator vs. distance to the next eigenvalue) controls speed, not the raw distance.

There is no closed-form radical formula for $n\ge 5$ (Characteristic Polynomial), but eigenvalues are still computable to any precision by iterative methods like these or the QR Algorithm.

Plain power method keeps returning ${\lambda }_1$ — the dominant direction re-dominates. To reach ${\lambda }_2$ you must deflate (remove the $v_1$ component) or use a shift/inverse strategy.

Only valid when $x$ is already normalized ($x^{\top }x=1$). In general $\mu =\frac{x^{\top }Ax}{x^{\top }x}$; dropping the denominator on an unnormalized vector rescales your eigenvalue estimate.

Factoring the largest term leaves ratios $({\lambda }_i/{\lambda }_1{)}^k$ that are all $<1$ and vanish; factoring anything smaller would leave a term $\ge 1$ that blows up, hiding the convergence.

$A^{-1}$'s eigenvalues are $1/{\lambda }_i$, so $A$'s smallest $|\lambda |$ becomes the largest $|1/\lambda |$ — the one power method naturally amplifies. See Convergence Rates.

${\lambda }_j-\sigma$ becomes tiny, so $\frac{1}{{\lambda }_j-\sigma }$ becomes enormous and dwarfs every other reciprocal — the ratio to the next term collapses toward $0$, so one or two steps can suffice.

The Rayleigh quotient is the least-squares best eigenvalue over all directions of the current vector, so it averages out the not-yet-decayed error components; a single component is corrupted by whichever non-dominant piece happens to sit there.

The matrix $A-\sigma I$ is the same at every iteration; only the right-hand side $x_k$ changes. One $O(n^3)$ factorization plus cheap $O(n^2)$ forward/back-substitutions per step beats repeated full solves.

The non-dominant error decays as $|{\lambda }_2/{\lambda }_1{|}^k$; if that ratio $\approx 0.99$, each step removes only $1\%$ of the error, so you need hundreds of iterations to converge.

In exact arithmetic $A^k{x}_0={\lambda }_2^k{v}_2$ forever — you converge to the wrong eigenvector. In floating point, rounding usually re-injects a tiny $v_1$ component that eventually dominates.

$A$ is singular, so $A^{-1}$ (or the solve $Ay=x$) does not exist — inverse iteration cannot start. This is fine for shifted inverse iteration as long as $A-\sigma I$ is nonsingular.

Then $|{\lambda }_1|=|{\lambda }_2|$ with the two complex conjugates tied in magnitude, so the real power method cannot pick a single real eigenvector and instead oscillates — a signal to switch to the QR Algorithm.

The two nearest eigenvalues are equidistant, so $|{\lambda }_j-\sigma |=|{\lambda }_m-\sigma |$ and the convergence ratio is $1$ — shifted inverse iteration stalls; nudge $\sigma$ toward the one you actually want.

For symmetric $A$ the Rayleigh quotient error is $O({\text{eigenvector error}}^2)$, so ${\mu }_k$ is far more accurate than the vector itself and converges roughly twice as fast in exponent.

The iterate still aligns with the $v_1$ direction (up to sign); $-v_1$ is just as valid an eigenvector as $v_1$, and the Rayleigh quotient returns the same ${\lambda }_1$ regardless of sign.