Hautus lemma¶

A rank-test lemma characterizing controllability, observability, stabilizability, and detectability of linear time-invariant state-space systems at eigenvalues.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4834
Domain group: Interdisciplinary & Synthetic
Origin domain: Systems Thinking & Cybernetics
Subdomain: Linear Control Theory → Systems Thinking & Cybernetics

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.

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Check Every Wobble

Imagine a toy that wiggles in a few special ways all by itself. You have buttons to push it. The Hautus lemma is a check: for every special wiggle, do your buttons have any effect on it? If even one wiggle ignores every button, you can't fully steer the toy.

Mode-by-Mode Steering Test

Engineers describe some machines with matrices: one says how the machine moves on its own, and another says how your controls push it. Each machine has natural patterns of motion called modes. If some mode is completely untouched by your controls, you can't fully steer the machine. The Hautus lemma gives a test: for each mode, check whether a certain combined table of numbers has "full rank." If every mode passes, the machine can be steered. A similar test checks whether your sensors can see every mode.

PBH Eigenvalue Rank Test

A linear system is written x' = Ax + Bu with output y = Cx. The eigenvalues λ of A correspond to the system's natural modes. The Hautus lemma, also called the Popov–Belevitch–Hautus (PBH) test, says the system is controllable exactly when the block matrix [λI − A, B] has full rank n for every eigenvalue λ. Similarly, it is observable exactly when the matrix formed by stacking λI − A on top of C has full rank. When the rank drops, there is an eigenvector that B or C completely ignores, which is a mode the inputs can't reach or the sensors can't see. The usual controllability test checks one big matrix, while this test checks mode by mode.

For the linear system x' = Ax + Bu, y = Cx, with A an n×n matrix over a specified field, the PBH (Popov–Belevitch–Hautus) tests express structural properties as rank conditions. The pair (A,B) is controllable if and only if rank[λI − A, B] = n for every eigenvalue λ of A. Rank can only drop at eigenvalues, so checking those is enough. Dually, (A,C) is observable if and only if the stacked matrix [λI − A; C] has rank n. The mechanism is eigenvector-based. An uncontrollable mode gives a left eigenvector w with wᵀA = λwᵀ and wᵀB = 0. An unobservable mode gives a right eigenvector v with Av = λv and Cv = 0. Ruling out such vectors is exactly the full-rank condition. Related properties come from restricting the range of λ to the relevant region; for example, stabilizability restricts to eigenvalues with non-negative real part. In every case, the field, the eigenvalue range, and the property claimed must match.

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.
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) — outgoing typed relations

Hautus lemma is a kind of Verification Prime

What makes it its own entry: the domain-specific identity determined by the system matrices, field, eigenvalue range, and exact full-rank condition corresponding to the claimed control property are stated and equivalent.

Hierarchy path (1) — routes to 1 parentless root

Hautus lemma → is a kind of Verification → is a kind of Evaluation → presupposes Comparison

Neighborhood in Abstraction Space¶

Hautus lemma sits in a crowded region of the domain-specific corpus (17^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Linear Control & Circuit Systems (12 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25