Hautus lemma¶
A rank-test lemma characterizing controllability, observability, stabilizability, and detectability of linear time-invariant state-space systems at eigenvalues.
Core Idea¶
The Popov–Belevitch–Hautus tests state system properties through full-rank block matrices such as [λI−A,B] or their observability transpose forms. An uncontrollable or unobservable eigenmode produces a left or right eigenvector annihilated by the input or output map; excluding such vectors is equivalent to the rank condition. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Hautus lemma belongs to linear control theory and is useful where the analyst can specify the typed linear control theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent. The scope is broad within that domain but bounded by the need for the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent. Conceptual linear-systems theorem only; safety-critical control design requires validated models and qualified engineering review.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Hautus lemma can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Hautus lemma. Hautus lemma compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed linear control theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of linear control theory because they reuse the typed linear control theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, An uncontrollable or unobservable eigenmode produces a left or right eigenvector annihilated by the input or output map; excluding such vectors is equivalent to the rank condition., and type the carrier, state every parameter and convention in the definition, test that the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Hautus lemma Domain-specific
Parents (1) — more general patterns this builds on
-
Hautus lemma is a kind of Verification Prime
The proposed strict upward parent is
prime:verification.
Hierarchy path (1) — routes to 1 parentless root
- Hautus lemma → Verification → Evaluation → Comparison → Self Checking
Neighborhood in Abstraction Space¶
Hautus lemma sits in a crowded region of the domain-specific corpus (30th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Feedback Control & Dynamical Systems (29 abstractions)
Nearest neighbors
- Kalman–Yakubovich–Popov lemma — 0.92
- Separation principle — 0.91
- Full state feedback — 0.90
- Linear complex structure — 0.90
- State-transition matrix — 0.90
Computed from structural-signature embeddings · 2026-09-08