The Operator View

A design matrix does more than hold samples. Once it is used to form correlations, covariances, kernels, or transitions, it defines an operator-like object: something that maps one vector representation into another or scores how directions in the data relate.

That is why the same language appears in statistics, control, signal processing, quantum mechanics, and numerical methods. The details differ, but the reusable structure is linear transformation on a vector space, often with an inner product that makes angles, projections, and norms meaningful.

Why Eigenvalues Show Up

Eigenvalues and eigenvectors are not decorative math. They identify directions that the operator preserves up to scaling. In a covariance matrix, that can reveal dominant variance directions. In a transition operator, it can reveal stable or slowly changing modes. In a physical operator, it can reveal measurable states or conserved structure.

The useful connection is practical: once a problem is represented as vectors plus operators, decomposition methods become a way to inspect the model's internal shape.

The Limit

The phrase "Hilbert space" can be too broad if it is used loosely. In day-to-day modeling, the starting point is often finite matrices and finite feature vectors. The infinite-dimensional language is valuable when the problem truly needs function spaces, kernels, or limiting arguments.