Hilbert-Space Operators as Statistical Models
August 1, 2023
Hilbert-space operators become concrete when data supplies the state vectors and correlation structure. Design matrices full of cross-correlation or mutual information can define transformations whose eigenvalues reveal the model’s internal modes.
Brian Greenforest connects statistical modeling, operator theory, quantum physics, control theory, tensors, and general relativity through this shared linear-algebraic structure.
Build the Operator From Relationships
A state vector records coordinates in a chosen basis. A Hermitian matrix can act as an operator on that state, preserving real eigenvalues and an orthogonal set of eigenvectors under the right conditions.
Correlation and information measurements populate the matrix. The resulting eigensystem identifies directions that the transformation amplifies, suppresses, preserves, or separates.
Use One Language Across Disciplines
Quantum observables, covariance analysis, control modes, and geometric tensors all use related operator machinery. Seeing the common structure makes each domain easier to connect and implement.
Researchers and educators can map a real design matrix into operator language, interpret its eigensystem, and show where statistical measurements become physical or computational modes.
LinkedIn status when archived: Visible to anyone on or off LinkedIn.
If we have two states in vector representation on infinitely-dimensional Hilbert space, and we're trying to trace the transition probabilities from one such state to another, this is what all this operator theory is about. Collecting tons of cross-correlation or mutual information data in your design matrices seem to help building statistical models that can be represented as Hermitian operators acting on the vectors in Hilbert space. When #eigenvalues are mentioned in all kinds of contexts, these apply to these transformational operators, and study the operators' inner structure. Why so little literature mentions how all these fundamental concepts relate and how we apply them, at the places where we use them?
#matrices #tensors #operators #controltheory #quantumphysics #generalrelativity