Eigendecomposition Workflow¶
Computational method — instantiates Invariant-Mode Decomposition Design
Takes an explicitly known linear operator and returns its complete set of invariant directions together with the scalar gain of each — the full modal picture the rest of the analysis reads from.
Everything else in this archetype needs a set of modes to work on; Eigendecomposition Workflow is the step that produces them. Given a linear operator you can write down explicitly — a transition matrix, a Jacobian, a coupling matrix — it solves for every invariant direction the operator preserves (its eigenvectors) and the scalar factor by which each is stretched or damped (its eigenvalues), returning the complete basis and spectrum in one pass. Its defining trait is exactly that completeness-from-an-explicit-operator: it does not pick a favourite mode, classify anything, or measure a real system — it factors a known transformation into the independent directions that make its tangled action simple. If you only wanted the single loudest mode you would reach for Power-Iteration Probe; if you had no operator to write down you would reach for Mode-Shape Testing. This is the full analytic decomposition that both of those approximate or sidestep.
Example¶
A conservation team models an age-structured animal population with a Leslie matrix — an explicit operator whose rows encode each age class's survival and fecundity. Multiplying a population-by-age vector by the matrix advances it one year, and the team wants to understand the whole behaviour, not one summary number. Eigendecomposition of the matrix returns the full modal picture: the dominant eigenvalue (≈1.03) is the long-run annual growth rate, and its eigenvector is the stable age distribution the population settles into. But the workflow also returns the subdominant, complex eigenvalues — the damped oscillatory modes that explain the transient cohort waves ("baby booms" echoing through the age structure for years before they fade). That is what a full decomposition buys over a dominant-mode shortcut: not just "the population grows at 3%," but "and here is the wobble on the way there, how big it is, and how fast it settles." Every mode is kept traceable back to real age classes, so the ecologists can read a mode as a story about cohorts rather than as an abstract vector.
How it works¶
What sets this apart from its siblings is that it factors the whole explicit operator:
- Scope and assemble the operator. Fix the transformation and the coordinates it acts in, and write it out explicitly as a matrix in those coordinates.
- Solve for all eigenpairs. Solve
T v = λ vfor the entire set of eigenvectors and eigenvalues — not a leading subset — using a symmetric solver when the operator is symmetric/normal and a general non-symmetric solver otherwise. - Return basis + spectrum, sorted. Deliver the eigenvectors (the invariant-mode basis) alongside their eigenvalues (the gain spectrum), ordered by whatever convention the analysis uses.
- Preserve traceability. Keep the change-of-basis that maps modal coordinates back to the original named variables, so a mode is interpretable rather than a bare direction.
Tuning parameters¶
The dials that adapt the workflow to a specific operator:
- Coordinate framing — the basis the operator is assembled in. A better-conditioned framing (e.g. non-dimensionalised, or mass-normalised) yields cleaner, more separable, more interpretable modes.
- Symmetric vs. general solver — exploiting symmetry gives real, orthogonal modes cheaply; a general operator needs a heavier solver and may return complex, non-orthogonal modes that are harder to read.
- Full vs. banded spectrum — every eigenpair, or only a band of interest. Full is the point of this mechanism, but a very large operator can force truncation — at which point Power-Iteration Probe or Singular Value Decomposition is the better tool.
- Sorting & normalisation convention — ordering by magnitude, real part, or variance, and how eigenvectors are scaled; this quietly defines what "dominant" and "mode 1" mean for every downstream mechanism.
When it helps, and when it misleads¶
Its strength is decisive: when the operator is genuinely known and well-behaved, it converts a coupled, tangled transformation into a set of directions that each evolve independently — the exact structure the whole archetype is built to exploit, delivered without approximation.
Its failure mode is subtler than "wrong arithmetic." For non-normal operators the eigenvectors can be nearly parallel and wildly ill-conditioned, so the tidy modal picture is numerically fragile, and — worse — the eigenvalues alone can hide large transient growth that erupts before the asymptotic decay the spectrum promises.[1] The classic misuse is decomposing an operator that isn't actually linear or stationary across the range of interest and then treating its modes as real, permanent structure. The discipline that guards against this is to check the conditioning and orthogonality of the eigenvectors, reach for Singular Value Decomposition when the operator is strongly non-normal, and remember that a clean spectrum is a statement about eventual behaviour, not about the path there.
How it implements the components¶
Eigendecomposition Workflow fills the compute core of the archetype — the components a decomposition method itself produces:
transformation_scope— it pins down the explicit operator and the coordinates it acts in; scoping the transformation is its first step.invariant_mode_basis— its primary output: the complete set of eigenvectors, the invariant directions.modal_gain_spectrum— its co-output: the eigenvalue attached to each mode, the scalar gain.
It computes the raw decomposition but does not act on it — which modes count as dominant is Power-Iteration Probe's dominant_mode_selection_rule, the stable/unstable verdict is Modal Stability Analysis's stable_unstable_mode_partition, and how well a truncated set reconstructs the whole is Residual Reconstruction Test's reconstruction_residual_check.
Related¶
- Instantiates: Invariant-Mode Decomposition Design — it supplies the invariant modes and gains the rest of the appraisal operates on.
- Sibling mechanisms: Power-Iteration Probe · Singular Value Decomposition · Spectral Decomposition Report · Modal Stability Analysis · Mode-Shape Testing · Principal Component Analysis · Reduced-Order Model
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Eigendecomposition Workflow operates as a computation, comparison, model, or analytic representation used to infer, estimate, or choose because it takes an explicitly known linear operator and returns its complete set of invariant directions together with the scalar gain of each — the full modal picture the rest of the analysis reads from.
Independent corroboration: The frozen evidence defines Eigendecomposition Workflow as 'Takes an explicitly known linear operator and returns its complete set of invariant directions together with the scalar gain of each — the full modal picture the rest of the analysis reads from', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Single lineage
Present-day reach: Multi-domain
Rationale: Linear algebra cohered eigendecomposition as deriving an operator's invariant directions and corresponding scalar gains.
Related originating lineages:
- Physics — Normal-mode analysis made the decomposition a standard method for understanding coupled physical systems.
Review resolution: The current reviewers agree that mathematics is primary. For the reported differences (alternate_origin_disagreement), the evidence supports single_lineage, multi_domain, and physics; these choices preserve materially formative origins without conflating later domain reach.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
The decomposition is only as useful as the traceability it carries. The archetype's standing requirement is that the modal model stay mappable back to the original variables; if the change-of-basis is discarded, you are left with mathematically correct modes that no one can act on because no one can say what, in real terms, each mode is. Keeping the mapping is what lets a downstream sibling name a mode "the cohort-echo mode" rather than "eigenvector 3."
References¶
[1] Trefethen, L. N., & Embree, M. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press (2005). Shows that nonnormal systems can have ill-conditioned eigenvectors and large transient amplification even when their eigenvalues predict eventual decay. registry ↩