Spread of a matrix¶
The maximum complex-plane distance between any two eigenvalues of a square matrix.
Core Idea¶
Spread depends only on the spectrum but not eigenvectors, repeated single eigenvalues give zero even for a non-scalar defective matrix and it differs from spectral radius and singular-value spread. All eigenvalue pairs are compared by complex modulus; the diameter of the finite spectral set is selected as one nonnegative measure of spectral dispersion. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Spread of a matrix belongs to matrix analysis and is useful where the analyst can specify the typed matrix analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the square real or complex matrix, eigenvalues with or without multiplicity, spectral set in the complex plane, pairwise distances absolute lambda_i minus lambda_j, maximum or diameter definition, zero-spread cases, invariance under similarity and scalar translation, scaling behavior and bounds using norms numerical range trace or Hermitian parts are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the square real or complex matrix, eigenvalues with or without multiplicity, spectral set in the complex plane, pairwise distances absolute lambda_i minus lambda_j, maximum or diameter definition, zero-spread cases, invariance under similarity and scalar translation, scaling behavior and bounds using norms numerical range trace or Hermitian parts are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Spread of a matrix. Spread of a matrix compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed matrix analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the square real or complex matrix, eigenvalues with or without multiplicity, spectral set in the complex plane, pairwise distances absolute lambda_i minus lambda_j, maximum or diameter definition, zero-spread cases, invariance under similarity and scalar translation, scaling behavior and bounds using norms numerical range trace or Hermitian parts are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of matrix analysis because they reuse the typed matrix analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, All eigenvalue pairs are compared by complex modulus; the diameter of the finite spectral set is selected as one nonnegative measure of spectral dispersion., and type the carrier, state every parameter and convention in the definition, test that the square real or complex matrix, eigenvalues with or without multiplicity, spectral set in the complex plane, pairwise distances absolute lambda_i minus lambda_j, maximum or diameter definition, zero-spread cases, invariance under similarity and scalar translation, scaling behavior and bounds using norms numerical range trace or Hermitian parts are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Spread of a matrix Domain-specific
Parents (1) — more general patterns this builds on
-
Spread of a matrix is a kind of Measurement Prime
The proposed strict upward parent is
prime:measurement.
Hierarchy path (1) — routes to 1 parentless root
- Spread of a matrix → Measurement
Neighborhood in Abstraction Space¶
Spread of a matrix sits in a crowded region of the domain-specific corpus (22nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Matrix Structure & Linear Maps (48 abstractions)
Nearest neighbors
- M-matrix — 0.92
- Orthostochastic matrix — 0.91
- Modal matrix — 0.91
- Defective matrix — 0.91
- Complex Hadamard matrix — 0.91
Computed from structural-signature embeddings · 2026-09-08