Extract spectral information from repeated actions

Try

Apply A=diag(3,1) repeatedly to x₀=(1,1). The results are (3,1),(9,1),(27,1). The first direction increasingly dominates. Normalize after every multiplication to avoid uncontrolled growth; normalization changes scale but not the direction produced by exact arithmetic. This is the power method. Its success comes from a spectral gap and a nonzero initial component in the dominant direction.

Predict

Predict what happens from x₀=(0,1). The dominant component is absent and remains absent exactly. Also consider diag(1,−1): equal eigenvalue magnitudes can produce alternation rather than one selected direction. A negative dominant eigenvalue can flip the sign of normalized vectors even when the underlying line converges. Define what convergence means before plotting coordinate differences or declaring failure.

Experiment

For the diagonal example, inspect component ratios.

Number of powers	Unnormalized vector	Subdominant/dominant ratio
0	(1,1)	1
1	(3,1)	1/3
2	(9,1)	1/9

Replace the second eigenvalue by 2.9 and predict the much slower reduction factor. This demonstrates a gap-dependent rate, not a universal number of iterations.

Understand

Assume A is diagonalizable, with |λ₁|>|λ₂|≥⋯ and x₀=Σcⱼvⱼ with c₁≠0. Dividing A^kx₀ by λ₁^k leaves c₁v₁ plus terms multiplied by (λⱼ/λ₁)^k, which vanish. This proves direction convergence, up to sign or phase, under the stated assumptions. Nonnormal eigenvectors can be poorly conditioned, influencing transient behavior and constants. Defective matrices require additional Jordan analysis; the elementary proof should not silently be extended to every matrix. A Rayleigh quotient can estimate an eigenvalue, but its interpretation is strongest for self-adjoint matrices.

Apply

Shifted inverse iteration applies (A−μI)⁻¹, whose eigenvalues are 1/(λⱼ−μ), provided the shift is not an eigenvalue. The eigenvalue nearest μ can become dominant in magnitude. Implement this by solving a linear system at each step, reusing a factorization for a fixed shift. Do not explicitly form an inverse. To check a candidate unit vector x and value θ, compute residual r=Ax−θx. For self-adjoint A, expansion in an orthonormal eigenbasis proves dist(θ,spectrum A)≤‖r‖. The same simple bound is not valid for arbitrary nonnormal matrices; small residuals there can reflect pseudospectral sensitivity.

For a self-adjoint matrix, the Rayleigh quotient θ=⟨Ax,x⟩ for unit x makes the residual orthogonal to x. This follows by subtracting θ⟨x,x⟩ from ⟨Ax,x⟩. It is an internal consistency check, not a proof that x is close to one isolated eigenvector. If two eigenvalues are nearly equal, many mixtures have small residuals relative to their separation. Eigenvector accuracy then needs gap information in addition to the residual norm.

Continue

An eigenvalue residual does not by itself certify a unique eigenvector when eigenvalues cluster. Report the gap assumptions, start, normalization, residual and stopping reason. Test on a diagonal matrix, a negative dominant eigenvalue, equal magnitudes and a start missing the target component. Large-scale eigenproblems need richer methods such as QR or Krylov iterations beyond this introduction. Next, time-stepping algorithms apply repeated updates to differential equations, where amplification and accumulated local error interact over a finite interval.

Review calculus and linear algebra when a prerequisite needs practice. Continue to Turn an ODE slope into a controlled first-order step.