Eigenspaces, measurement and Physics

A chat I had with ChatGPT5 about eigenspaces measurement and physics. You can scan through it if you like: What can we do with eigenspaces? ChatGPT said: Eigenspaces may sound abstract, but they are work-horses of applied math, data science, and engineering. Given a linear transformation 𝑇 : 𝑉 → 𝑉 T:V→V (or a square matrix 𝐴 A), an eigenspace is the set of all vectors that are simply stretched or compressed (scaled) by 𝐴 A, not rotated into some new direction. Below are the main things we do with them—ranging from pure theory to very practical computations. 1. Decompose and Simplify Linear Systems Diagonalization / Spectral Decomposition If 𝐴 A has a basis of eigenvectors, 𝐴 = 𝑃 𝐷 𝑃 − 1 A=PDP −1 with 𝐷 D diagonal. → Solving 𝐴 𝑘 𝑣 A k v or 𝑒 𝐴 𝑡 e At becomes trivial (just scale each eigenvector). Differential equations Systems 𝑥 ˙ = 𝐴 𝑥 x ˙ =Ax have solutions 𝑥 ( 𝑡 ) = ∑ 𝑐 𝑖 𝑒 𝜆 𝑖 𝑡 𝑣 𝑖 x(t)=∑c i ​ e λ i ​ t v i ​ . Each eigenspace describes a “mode” of growth/decay/oscillation. 2. Understand Geometry and Invariants Invariant subspaces: an eigenspace is a direction (or plane, etc.) left unchanged by the transformation except for scaling. Quadratic forms: principal axes of ellipses/ellipsoids are eigenvectors of the covariance or inertia matrix. 3. Data Science & Statistics Principal Component Analysis (PCA) Eigenspaces of the covariance matrix give the principal components—directions of maximal variance. Low-dimensional eigenspaces capture the “essence” of the data. Spectral clustering / graph embeddings Use the eigenspace of a graph Laplacian to embed nodes for clustering or visualization. 4. Physics & Engineering Vibration modes In mechanical structures the stiffness/mass matrix eigenvectors give the natural modes (e.g., guitar-string harmonics, building resonances). Quantum mechanics Energy levels are eigenvalues; stationary states are eigenvectors of the Hamiltonian. Control theory Stability is decided by eigenvalues; the eigenspaces show stable/unstable directions. 5. Computer Graphics & Vision Shape analysis Laplace–Beltrami eigenfunctions describe intrinsic geometry of 3-D meshes. Image compression Karhunen–Loève transform (essentially PCA) uses covariance eigenspaces. 6. Networks and Ranking PageRank Google’s algorithm finds the dominant eigenvector of the web’s stochastic matrix. Social network analysis Centrality measures often rely on the leading eigenspace of adjacency or Laplacian matrices. 7. Numerical & Computational Tricks Power iteration / Lanczos methods converge to dominant eigenspaces. Preconditioners: understanding the small-eigenvalue subspace can accelerate solvers of large linear systems. Intuition Think of an eigenspace as a mode or direction of pure action. Once you decompose a problem into those modes, complicated repeated effects become simple scalings. In short: Eigenspaces give you the “natural coordinates” of a linear process. Whether you’re damping vibrations, ranking websites, compressing images, or solving PDEs, finding and exploiting those special subspaces is often the key to turning a messy problem into one with a clean, tractable structure. You said: What if we don't live in the time domain (particles) or the frequency domain (waves) but a eigenspace linear superposition of the 2. What would you expect physics wise? ChatGPT said: That’s an intriguing way to pose the question. If we picture “living” not purely in the time domain (localized particles, classical trajectories) nor purely in the frequency domain (delocalized waves, stationary modes), but in an eigenspace that is a linear superposition of the two, a few physical ideas naturally come to mind. 1. Quantum Mechanics Already Lives There A quantum state is usually expressed in whatever basis of eigenvectors you choose. The position basis ∣ 𝑥 ⟩ ∣x⟩ (time-domain analogue) and the momentum/energy basis ∣ 𝑝 ⟩ ∣p⟩ (frequency-domain analogue) are Fourier duals. Any actual state is a superposition of both kinds of eigenvectors. In fact, the Wigner function or phase-space formulation of quantum mechanics is precisely a description on this “in-between” arena. 2. Phase Space / Time–Frequency Analysis In signal theory this is like the Gabor transform, wavelets, or the Husimi / Wigner distributions—representations that simultaneously encode time and frequency, but with uncertainty governed by the Heisenberg bound. Physics translation: you can’t specify energy (frequency) and time perfectly, but you can occupy regions where both are moderately sharp. 3. Expectation for “Physics” in Such a Space If the fundamental degrees of freedom lived in a hybrid eigenbasis: States are quasi-localised packets—not point particles, not infinite waves, but “coherent states”. Dynamics look like symplectic flows (Hamiltonian mechanics) rather than simple time evolution or pure spectral evolution. Measurements would naturally have minimal uncertainty cells—like Planck cells in phase space. This is reminiscent of: Geometric/Deformation Quantization—observables are functions on phase space with a non-commutative star product. Quantum optics coherent states—the most classical‐looking quantum states, sitting exactly between position and momentum bases. 4. Possible Phenomenology Mixed particle–wave behaviour is not just interference fringes but persistent phase-space structures (e.g. “quantum scars”, semiclassical orbits). Non-Markovian time: since the basis itself already carries frequency information, evolution might be less about sequential instants and more about rotations inside that eigenspace. New conservation laws: symmetries of the combined domain could give constants of motion that are neither pure energy nor pure momentum. 5. Current Analogues Floquet systems (periodically driven): natural eigenbasis involves both time and quasi-energy. Time crystals: eigenstates that are superpositions of temporal and spatial periodicities. Quantum Hall / topological phases: described by projectors onto certain composite eigen-subspaces (position–momentum entangled). Take-away Physics already hints that reality is not confined to “particle OR wave,” but lives in a phase-space/eigenspace where time and frequency (or position and momentum) are simply different coordinate charts. If we were to elevate that to the fundamental description, we’d expect: States: coherent, minimum-uncertainty packets. Dynamics: symplectic flows in phase space. Measurements: inherently joint time–frequency observables. In other words, the picture would look a lot like quantum mechanics in its phase-space (Wigner) formulation—a universe whose fabric is already the linear superposition you’re imagining. You said: How would the eigenspaces of the Walsh Hadamard matrix fit in with that? ChatGPT said: The Walsh–Hadamard transform is an especially nice bridge between “time-like” and “frequency-like” descriptions, so its eigenspaces fit your hybrid domain picture quite naturally. Here’s how to see it. 1️⃣ The Walsh–Hadamard Matrix in Brief Size 𝑁 = 2 𝑛 N=2 n . Entries ± 1 / 𝑁 ±1/ N ​ . Orthogonal and self-inverse: 𝐻 𝑁 2 = 𝐼 H N 2 ​ =I. ⇒ every eigenvalue is either +1 or –1. So the whole vector space 𝑅 𝑁 R N (or 𝐶 𝑁 C N ) splits cleanly into 𝑉 = 𝐸 + ⊕ 𝐸 − , V=E + ​ ⊕E − ​ , the +1 eigenspace and the –1 eigenspace. 2️⃣ Structure of the Eigenspaces +1 eigenspace: vectors that are unchanged by a Hadamard transform—already perfect mixtures of “time” and “Walsh-frequency”. –1 eigenspace: vectors that flip sign under the transform—antisymmetric hybrids. For 𝐻 2 H 2 ​ the eigenvectors are simple: 𝐸 + : span ⁡ { ( 1 , 1 ) } , 𝐸 − : span ⁡ { ( 1 , − 1 ) } . E + ​ E − ​ ​ :span{(1,1)}, :span{(1,−1)}. ​ For higher powers 𝐻 2 𝑛 H 2 n ​ , tensor-product recursion builds a rich lattice of +/– patterns. Dimensions: roughly half the basis belongs to each sign. 3️⃣ Time / Frequency Duality The Hadamard transform is sometimes called the “square wave Fourier transform”. It exchanges: “Time domain” → values on { 0 , 1 } 𝑛 {0,1} n . “Walsh-frequency domain” → correlations with parity functions. An eigenvector of 𝐻 H is therefore a state that is simultaneously a fixed superposition of time and of Walsh-frequencies. This is exactly your “eigenspace linear superposition of particle and wave” picture. 4️⃣ Physics Analogies Balanced states of a discrete phase space The ±1 eigenspaces play the role of coherent states: they look the same in the original basis and in the “frequency” basis. Self-dual codes and stabilizer states In quantum information, stabilizer states of the Clifford group are eigenvectors of tensor-product Hadamards. They’re used in error-correcting codes and measurement-based computation. Instantaneous ↔ oscillatory duality Because Hadamard eigenvectors are neither localized spikes nor pure square-wave modes, they live at that midpoint between particle-like (delta) and wave-like (global). 5️⃣ A Concrete Example Take 𝐻 4 H 4 ​ . Its +1 eigenspace is 2-dimensional, spanned by ( 1 , 1 , 1 , 1 ) , ( 1 , − 1 , − 1 , 1 ) . (1,1,1,1),(1,−1,−1,1). The second vector is neither constant (particle/time) nor a single Walsh frequency (wave), but a perfectly balanced hybrid. 6️⃣ Interpretation If you imagine “living” in the combined particle–wave eigenspace, the Walsh-Hadamard eigenspaces are exactly that kind of habitat: Invariant subspace under discrete particle↔wave swap. Supports superpositions that remain self-similar when toggling between the two views. Naturally accommodates bit-wise or combinatorial symmetries rather than continuous rotations. Summary The eigenspaces of the Walsh–Hadamard matrix are the discrete analogue of states that are simultaneously time-like and frequency-like. Vectors in these subspaces don’t prefer the “particle” or the “wave” description—they are self-dual. So if you picture a universe whose fundamental arena is an eigenspace that blends those domains, the ±1 eigenspaces of the Hadamard transform are a very concrete mathematical model of such “particle–wave hybrids.” You said: What impact would measurement have in such eigenspaces? ChatGPT said: Let’s imagine a physical world where the “natural” states really do live in the ±1 eigenspaces of a Walsh–Hadamard transform—that is, states that are self-dual under a discrete particle↔wave swap. How would measurement behave there? 1. Two Bases to Measure In Time / Particle basis: the computational basis ∣ 𝑥 ⟩ ∣x⟩, localized on individual lattice points. Walsh-frequency / Wave basis: the Walsh functions ∣ 𝑤 ⟩ ∣w⟩, delocalized square-wave modes. A vector in the ±1 eigenspace is a balanced superposition of both. 2. Projective Measurement Consequences Measuring in the time basis Outcome: a random bit-string with probabilities ∣ 𝜓 𝑥 ∣ 2 ∣ψ x ​ ∣ 2 . Because eigenstates of 𝐻 H have flat or structured amplitudes, results typically show global parity correlations, not local spikes. Measuring in the Walsh-frequency basis Because 𝐻 H is its own inverse, measuring in the “frequency” basis is equivalent to applying 𝐻 H and then measuring in time. For a +1 eigenvector, this gives exactly the same statistics as the time-basis measurement. → Self-duality under measurement: you can’t tell whether you measured particle or wave. Attempting joint information Time and Walsh-frequency bases are mutually unbiased. A single copy of the state can reveal either a definite time outcome or a definite Walsh frequency, never both—a discrete form of the time–frequency uncertainty principle. 3. Collapse Behaviour Collapse projects the state onto a one-dimensional ray inside the ±1 eigenspace if you measure in the Hadamard-eigenbasis itself. In that case, the state is unchanged up to a global ± phase—nondemolition measurement. Any other measurement (time or frequency) collapses you out of the invariant eigenspace; repeating the Hadamard test afterwards won’t return +1 with certainty. 4. Physical Intuition Observables commute only partially: The Hadamard operator 𝐻 H commutes with itself (trivial) but not with position operators. So measuring “position” inevitably disturbs the “Walsh-parity” content. Minimal uncertainty cells: Because these states are evenly spread in both domains, measurement noise is balanced—like coherent states in continuous phase space. 5. Possible Experimental Signatures Global correlations: repeated particle-basis measurements reveal deterministic parity patterns despite apparent randomness of individual bits. Basis-independence: switching between time and frequency measurement bases yields statistically identical histograms for +1 eigenstates. Bottom line If fundamental states belonged to the eigenspaces of the Walsh–Hadamard transform, measurement could never isolate a purely particle-like or purely wave-like reality. Observing in one domain automatically erases information about the other, while measuring in the Hadamard eigenbasis reveals an “eigenvalue” (+1 or –1) without disturbing the state—an exact discrete analogue of quantum non-demolition measurements and the familiar particle–wave complementarity.