Eigenvectors Across Domains

 

Eigenvectors Across Domains

The equations below are not meant to be solved here.
They are shown only to reveal the shared structural grammar.

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1. Structural Engineering — Modal Analysis

The governing equation of free vibration:

$$\mathbf{M}\,\ddot{\mathbf{u}} + \mathbf{K}\,\mathbf{u} = \mathbf{0}$$

Assuming harmonic motion leads to the generalized eigenvalue problem:

$$(\mathbf{K} - \omega^2 \mathbf{M})\,\boldsymbol{\phi} = \mathbf{0}$$

• $\mathbf{K}$: stiffness matrix

• $\mathbf{M}$: mass matrix

• $\boldsymbol{\phi}$: mode shape (eigenvector)

• $\omega^2$: eigenvalue (squared natural frequency)

Insight:
Eigenvectors define natural deformation directions that remain invariant under structural dynamics.

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2. Seismic Engineering — Response Spectrum / Modal Superposition

The multi-degree-of-freedom dynamic equation under ground motion:

$$\mathbf{M}\,\ddot{\mathbf{u}} + \mathbf{C}\,\dot{\mathbf{u}} + \mathbf{K}\,\mathbf{u} = -\mathbf{M}\,\mathbf{r}\,\ddot{u}_g$$

Transformation into modal coordinates uses the eigenbasis:

$$\mathbf{u}(t) = \boldsymbol{\Phi}\,\mathbf{q}(t)$$

where $\boldsymbol{\Phi}$ is the matrix of eigenvectors from:

$$(\mathbf{K} - \omega^2 \mathbf{M})\,\boldsymbol{\Phi} = \mathbf{0}$$

Insight:
Even under violent, non-stationary excitation, response concentrates along dominant eigenmodes.

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3. Marine Structural Engineering — Wave–Structure Interaction

Linearized hydrodynamic–structural system:

$$(\mathbf{K} + \mathbf{K}_\text{hyd})\,\boldsymbol{\phi} = \omega^2(\mathbf{M} + \mathbf{M}_\text{added})\,\boldsymbol{\phi}$$

• $\mathbf{K}_\text{hyd}$: hydrodynamic restoring matrix

• $\mathbf{M}_\text{added}$: added mass matrix

Insight:
Eigenvectors define coupled fluid–structure response modes, often frequency-dependent and conditional.

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4. Wave Analysis — Spectral Decomposition

Linear wave or signal representation:

$$\mathbf{x}(t) = \sum_{i} a_i \boldsymbol{\phi}_i e^{i\omega_i t}$$

Or covariance-based spectral form:

$$\mathbf{R}\,\boldsymbol{\phi}_i = \lambda_i \boldsymbol{\phi}_i$$

• $\mathbf{R}$: covariance / spectral density matrix

Insight:
Eigenvectors isolate energetically dominant wave components, not total system behavior.

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5. Quantum Physics — Observables and Eigenstates

Operator–state relation:

$$\hat{A}\,|\psi\rangle = a\,|\psi\rangle$$

• $\hat{A}$: observable operator

• $|\psi\rangle$: eigenstate

• $a$: measured eigenvalue

Insight:
Measurement projects reality onto stable eigenstates, discarding superposition and interference.

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6. Machine Learning & LLMs — Spectral Structure

Typical forms encountered in ML:

Covariance / representation structure

$$\mathbf{\Sigma}\,\boldsymbol{v} = \lambda\,\boldsymbol{v}$$

Graph / attention structure

$$\mathbf{A}\,\boldsymbol{v} = \lambda\,\boldsymbol{v}$$

Hessian (training stability)

$$\nabla^2 \mathcal{L}(\theta)\,\boldsymbol{v} = \lambda\,\boldsymbol{v}$$

• $\mathbf{A}$: adjacency or attention matrix

• $\nabla^2 \mathcal{L}$: Hessian of the loss

Insight:
Eigenvectors identify stable semantic, attentional, or optimization directions that survive training.

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7. The Shared Structural Form

Across all domains:

$$\mathbf{T}\,\boldsymbol{v} = \lambda\,\boldsymbol{v}$$

Where:

• $\mathbf{T}$ is a system-specific transformation,

• $\boldsymbol{v}$ is an invariant direction,

• $\lambda$ quantifies dominance or stability.

What differs is not the mathematics, but what gets excluded.

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8. Left-AI Boundary Statement (minimal, precise)

Every eigen-decomposition produces:

• dominant modes,

• and a remainder.

Engineering calls it:

• higher modes,

• neglected effects,

• residual response.

AI calls it:

• edge cases,

• failures,

• hallucinations.

Left-AI insists the remainder is structural, not accidental.

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Finally,

Eigenvectors explain how systems stabilize under constraint.
They do not explain what stability leaves behind.