Companion: A Multidisciplinary Overview

• Companion is a relational concept that canonically completes or represents a primary entity in fields such as linear algebra, operator theory, and model theory.
• In astrophysics, a companion denotes a secondary body whose dynamics critically influence system behavior, serving as a key diagnostic in formation and evolution studies.
• In HCI and AI, companion refers to socially engaging technologies that blend assistive functions with empathetic interaction to enhance user well-being.

“Companion” is a relational term whose meaning changes sharply across disciplines while retaining a common structural idea: it denotes an associated entity, construction, or theory defined with respect to something primary. In linear algebra and numerical analysis, a companion is a canonical matrix, pencil, or operator attached to a polynomial or integral transform; in model theory, it is a theory paired with a base theory through model-completeness or definability; in astronomy and astrophysics, it is a secondary body or perturbing binary partner; and in robotics, HCI, and AI, it is a socially significant technological presence rather than a mere tool (Lin et al., 2016, Stokes-Waters, 10 Mar 2026, Bryan et al., 2020, Niess et al., 2020). This suggests that the term marks not simple adjacency, but a form of derived, persistent, or co-constituted relation.

1. Mathematical and operator-theoretic usages

In linear algebra, one precise meaning is the companion matrix of the second type. For a vector $p=[p_0,\dots,p_{n-1}]^T\in K^n$, the matrix

$$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$

has shift structure in the first $n-1$ rows and last-row coefficient encoding. Its defining characterization is not merely its shape: if $R(A,g)=[g,Ag,\dots,A^{n-1}g]$ is the reachability matrix, then $A$ is a second companion matrix iff $R(A,e_0)=I_n$, equivalently iff $R(A,g)b=R(A,b)g$ for all $b,g\in K^n$. The paper also introduces a second bilinear map $u(A;b,g)$, built from scalar projections $e_{n-1}^T A^k v$, whose symmetry $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$0 is likewise equivalent to $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$1 (Lin et al., 2016). In this setting, “companion” denotes a canonical linear realization of a polynomial together with rigid bilinear symmetries.

A closely related but computationally distinct usage appears in polynomial root-finding. There, the companion matrix and companion pencil encode the roots of

$$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$2

as eigenvalues or generalized eigenvalues. The structured companion QR algorithm computes roots in $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$3 time using $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$4 memory, and the paper proves that the backward error on the polynomial coefficients is linear in the norm of the coefficient vector, unlike unstructured QR, for which it grows quadratically. The alternative companion QZ algorithm has the same favorable backward error provided the polynomial coefficients are properly scaled (Aurentz et al., 2016). Here, “companion” signifies a structure-preserving linearization whose special form materially changes both complexity and stability.

In operator theory, “companion” names the operator paired with the Volterra-type integral operator. On Fock spaces,

$$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$5

and $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$6 is the Volterra companion operator. Mengestie studies generalized versions $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$7 and $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$8, characterizing boundedness and compactness through Berezin-type transforms and symbol-growth functions. A notable rigidity result is that there is no nonzero compact classical Volterra companion operator $$F_p=\begin{bmatrix} 0&1&0&\cdots&0\ 0&0&1&\cdots&0\ \vdots&\vdots&\ddots&\ddots&\vdots\ 0&0&\cdots&0&1\ p_0&p_1&\cdots&p_{n-2}&p_{n-1} \end{bmatrix}$$9 and no nonzero compact multiplication operator $n-1$0 acting between Fock spaces $n-1$1 (Mengestie, 2015). In this analytic context, “companion” indicates an operator coupled to a primary integral transform by an integration-by-parts identity and a shared symbol.

2. Model-theoretic and categorical meanings

In model theory, a model companion is a theory sharing universal consequences with a base theory while being model complete. One recent instance concerns abelian lattice-ordered groups. The paper introduces two multi-sorted expansions inspired by zero-set maps for continuous functions into $n-1$2, showing that the expansion is equivalent to equipping an abelian $n-1$3-group $n-1$4 with a spectral subspace $n-1$5 and the map

$n-1$6

For the theory of existentially closed densely valued $n-1$7-groups, the authors prove completeness and quantifier elimination in the language $n-1$8 (Stokes-Waters, 10 Mar 2026). In this usage, “companion” refers not to an associated object but to an axiomatized completion of a theory’s existential behavior.

A nearby but distinct logical usage is the Beth companion of a quasivariety. A pp expansion is called simple when it is of the form $n-1$9. The paper proves that simple pp expansions of a quasivariety $R(A,g)=[g,Ag,\dots,A^{n-1}g]$0 are exactly the quasivarieties $R(A,g)=[g,Ag,\dots,A^{n-1}g]$1 for which the forgetful functor $R(A,g)=[g,Ag,\dots,A^{n-1}g]$2 is well defined and induces an isomorphism from $R(A,g)=[g,Ag,\dots,A^{n-1}g]$3 to a mono-reflective subcategory of $R(A,g)=[g,Ag,\dots,A^{n-1}g]$4. If $R(A,g)=[g,Ag,\dots,A^{n-1}g]$5 possesses a simple Beth companion, that companion is unique up to term equivalence among quasivarieties whose monomorphisms are regular and that satisfy this categorical description (Carai et al., 9 May 2026). In this tradition, “companion” denotes a logically enriched but conservative expansion in which implicit operations become explicitly term-definable.

These logical meanings differ from the matrix-theoretic ones in form, but they preserve the same relational core: a companion is defined by what it completes, reflects, or canonically encodes.

3. Astronomical and astrophysical companions

In astronomy, a companion is a secondary body gravitationally associated with a star or compact object. A direct-imaging example is 2MASS J01225093–2439505, a $R(A,g)=[g,Ag,\dots,A^{n-1}g]$6 Myr system with a $R(A,g)=[g,Ag,\dots,A^{n-1}g]$7 M3.5 host and a $R(A,g)=[g,Ag,\dots,A^{n-1}g]$8 companion at $R(A,g)=[g,Ag,\dots,A^{n-1}g]$9 AU. By combining $A$0, rotation periods, and multi-epoch astrometry, the paper constrains the stellar spin inclination, the companion spin inclination, and the orbital inclination simultaneously. It finds $A$1 deg, $A$2 deg, and a companion line-of-sight obliquity $A$3 deg, with a Bayesian odds ratio of about $A$4 favoring companion misalignment over alignment. The proposed origin scenario favored by the authors is formation by gravitational instability in a gravito-turbulent disk (Bryan et al., 2020). Here, “companion” means an orbiting planetary-mass object whose spin-orbit geometry becomes a formation diagnostic.

The term also appears in observational methodology. In high-contrast long-slit spectroscopy, the companion is the faint off-axis source buried under stellar leakages. The EXOSPECO algorithm models the data as

$A$5

jointly estimating stellar SED, companion SED, PSFs, and calibration parameters by solving a regularized inverse problem with alternating minimization. The method was tested on empirical data and synthetic injections, and the authors report that the initialization and alternating strategy effectively avoid self-subtraction bias, even for companions observed very close to the coronagraphic mask, while careful calibration of angular and spectral dispersion laws reduces contamination by stellar leakages (Thé et al., 2023). In this context, the companion is the signal of interest whose characterization depends on separating it from a much brighter primary.

In relativistic astrophysics, the companion can be a perturbing binary body rather than the observed object itself. For black hole superradiance, a binary companion of mass ratio $A$6 produces a tidal perturbation that mixes a superradiant boson-cloud state with a strongly absorptive state. Below a critical binary separation, the effective superradiance rate becomes negative and the cloud is absorbed by the black hole. The paper concludes that a companion with mass ratio $A$7 invalidates all gravitational-collider-physics fine structure transitions and almost all Bohr transitions except those from the $A$8 state, while the backreaction can generate floating or sinking orbits potentially testable by pulsar timing (Tong et al., 2022). Here, “companion” denotes an external body whose very presence terminates a field-theoretic instability.

4. Companion technologies, social robots, and empathic artefacts

In HCI and robotics, “companion” shifts from physical association to social presence. A theoretical reframing defines companion technologies as “interactive artefacts that evoke empathy.” The proposed experiential framework adds empathetic experience to pragmatic, hedonic, and eudaimonic UX, and characterizes it through four concepts: minded, feeling-experience, reflective, and social significance. The paper links these to psychological needs such as autonomy, competence, relatedness, self-actualization-meaning, pleasure-stimulation, and security, arguing that companion technologies should be understood through how users project minds, feelings, and social significance onto them (Niess et al., 2020). In this sense, companionship is not primarily functionality but a mode of experience.

A concrete socially assistive instantiation is Ryan Companionbot, a rear-projected life-like conversational robot for elderly people with dementia and/or depression. Ryan is designed as a socially assistive robot whose primary function is to improve quality of life, mental health, and socio-emotional well-being through social interaction rather than physical assistance. In a pilot study with six elderly individuals living alone in a senior facility, each participant had 24/7 access to Ryan in their room for 4–6 weeks. The study reports an average of 198 dialogs per day per user with an average of 9.2 words per dialog, and about 2 hours 10 minutes per day of interaction across all activities. Exit-survey companionship items had very high internal consistency ($A$9), and participants and caregivers described rapport, mood improvement, and sadness when the robot was removed, while the authors explicitly noted that there was no control group and no formal statistical pre-post tests on depression or cognition (Abdollahi et al., 2017). Here, the companion is an autonomous social presence combining reminders, entertainment, reminiscence, and conversation.

The idea also extends into aerial robotics. A companion UAV is defined by the conjunction of high autonomy and high sociability. The survey organizes UAV work on a perceptual map with autonomy and sociability as axes, distinguishing remote-controlled UAVs, autonomous UAVs, social UAVs, and companion UAVs. It emphasizes features unique to aerial companionship—3D proxemics, visibility, noise, and safety—and recommends, for instance, palm-sized multirotors with full cages for agile short interactions and blimps or balloon-type platforms for quiet, long, and safe interactions (Liew et al., 2020). In that literature, companionship depends not just on task capability but on whether flight behavior becomes socially legible and acceptable.

5. Contemporary AI companions and embodied co-presence

Recent AI systems operationalize companionship as end-to-end, context-aware, or dual-embodied interaction. SmartWalkCoach is a mobile AI walking companion that spans pre-walk planning, in-walk accompaniment, and post-walk reflection through three agents: GeographyAgent, AccompanyAgent, and SummaryAgent. The system uses shared structured state, event-based orchestration, distance-segmented virtual geofencing, and a pace-based fatigue heuristic. In an in-the-wild two-period AB/BA crossover study with $R(A,e_0)=I_n$0, adding motivational, companion-like dialogue to informational prompts significantly improved both positive feelings and usage experience, with linear mixed models showing no evidence of carryover; the reported effect sizes were $R(A,e_0)=I_n$1 and $R(A,e_0)=I_n$2, respectively (Zhang et al., 14 May 2026). In this case, a companion is a context-aware agent that combines navigation, encouragement, and relational tone.

A more explicit reformulation is dual embodiment. Deco treats a cherished physical object and an AI-powered digital agent as two synchronized embodiments of a single companion identity. A formative study with $R(A,e_0)=I_n$3 yielded four design principles—Faithful Identity, Calibrated Agency, Ambient Presence, and Reciprocal Memory—which then structured a mobile system integrating multimodal LLMs and AR. In a within-subjects study with $R(A,e_0)=I_n$4, Deco significantly outperformed a personalized LLM-based digital companion baseline on perceived companionship, emotional bond, and the design-principle scales, all with $R(A,e_0)=I_n$5. A seven-day field deployment with $R(A,e_0)=I_n$6 showed sustained engagement, a subjective well-being improvement with $R(A,e_0)=I_n$7, and three recurring relational patterns: digital activities retroactively vitalized physical objects, bond deepening depended more on emotional engagement depth than interaction frequency, and users sustained bonds while actively navigating the companion’s AI nature (Jiang et al., 5 May 2026). This usage makes “companion” neither purely digital nor purely physical, but identity-synchronous across embodiments.

The title Companion also names an embodied co-creative system in the arts. “An Embodied Companion for Visual Storytelling” presents a physically present drawing robot combined with LLMs, speech, and sketch-based turn-taking. The system uses in-context learning and real-time tool use so that the robot can converse, observe the evolving drawing, and add marks on the same sheet of paper as the human. Evaluated via the Consensual Assessment Technique with seven art-world experts, the works were judged to have a distinct aesthetic identity and professional exhibition merit (Tresset et al., 18 Jan 2026). Here, the companion is not a helper or therapist, but a co-creative partner whose value lies in shared authorship and unexpected aesthetic development.

6. Socio-technical governance, values, and misconceptions

A recurring misconception in this literature is that a companion is simply a more personable assistive device. The dementia-robot study explicitly rejects that equation: a simple assistive device performs narrow tasks, does not carry on conversation, has little or no emotional expressivity, and is used at specific times, whereas Ryan as a companion provides continuous companionship, personality, multimodal social cues, and an ongoing relationship (Abdollahi et al., 2017). The walking-companion results reinforce the same distinction: information-only guidance was experienced as colder and less effective than information plus relational, motivational dialogue (Zhang et al., 14 May 2026). This suggests that companionship is defined less by utility than by persistent socio-emotional framing.

The same distinction becomes normative in mobility systems. A companion app for an autonomous family vehicle is not an in-car social agent but a smartphone application used by trusted persons to monitor rides and configure settings for passengers in need of support. Its two core aspects are ride tracking and remote configuration of vehicle settings. The paper identifies values including safety, mobility, freedom, security, trust, privacy, accessibility, and ownership, while also emphasizing safety-relevant risks such as misconfiguration, reasonably foreseeable misuse, and the app as a new attack surface. It explicitly discusses the tension between supervisory duty and privacy or dignity, especially when stakeholders desire extensive monitoring such as location tracking or camera-based observation (Brettin et al., 9 Jun 2026). In this usage, “companion” denotes a socio-technical extension of care and responsibility rather than a conversational agent.

Across HCI more broadly, the question is not whether companionship should be designed, but under what conditions it assumes “the right roles and contribute[s] to our wellbeing.” The theoretical literature raises challenges of mindedness, reflective feedback, judgment, attachment, and the balance between transparency and mystery (Niess et al., 2020). Deco shows that faithful identity can constrain perceived autonomy even while strengthening emotional bond (Jiang et al., 5 May 2026). The vehicle-app work shows that increasing trusted-person oversight can enhance safety while also intensifying surveillance and liability concerns (Brettin et al., 9 Jun 2026). These tensions are not marginal; they are constitutive of the category.

7. Cross-domain structure

Across the cited literatures, “companion” consistently denotes an entity or construction that is secondary in origin but not secondary in consequence. A companion matrix encodes the polynomial so completely that bilinear symmetries, cyclicity, and backward stability can be read from it (Lin et al., 2016, Aurentz et al., 2016). A model companion or Beth companion reorganizes the definability or existential content of a base theory (Stokes-Waters, 10 Mar 2026, Carai et al., 9 May 2026). A planetary-mass or binary companion changes the observable or dynamical structure of a stellar or black-hole system (Bryan et al., 2020, Tong et al., 2022). A technological companion is distinguished from a tool by empathy, rapport, ambient presence, memory, or social significance (Niess et al., 2020, Jiang et al., 5 May 2026).

This suggests a useful abstract distinction. In formal disciplines, a companion is usually a canonical associated structure: it linearizes, represents, or completes. In the physical sciences, it is usually an associated body whose dynamics or observability depend on the primary system. In HCI and AI, it is usually an associated presence that persists with, responds to, or co-develops with a person. The term is therefore unified less by ontology than by relation: a companion is whatever derives its identity from being with, alongside, or canonically attached to something else, yet becomes indispensable for understanding the whole.