Defective matrix¶
A square matrix lacking a full basis of eigenvectors and therefore not diagonalizable over the stated field.
Core Idea¶
Defectiveness depends on the field, repeated eigenvalues are necessary but not sufficient and generalized eigenvectors restore a Jordan basis without making the matrix diagonalizable. For at least one eigenvalue, geometric multiplicity is smaller than algebraic multiplicity, so eigenvectors cannot span the space and a nontrivial Jordan chain is required. 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¶
Defective matrix belongs to linear algebra and is useful where the analyst can specify the typed linear algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the square matrix and scalar field, characteristic polynomial and eigenvalues, algebraic and geometric multiplicities, eigenspace dimensions, missing eigenvector count, diagonalizability test, generalized eigenvectors and Jordan blocks and numerical sensitivity are explicit. The scope is broad within that domain but bounded by the need for the square matrix and scalar field, characteristic polynomial and eigenvalues, algebraic and geometric multiplicities, eigenspace dimensions, missing eigenvector count, diagonalizability test, generalized eigenvectors and Jordan blocks and numerical sensitivity are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the square matrix and scalar field, characteristic polynomial and eigenvalues, algebraic and geometric multiplicities, eigenspace dimensions, missing eigenvector count, diagonalizability test, generalized eigenvectors and Jordan blocks and numerical sensitivity 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 Defective matrix. Defective 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 linear algebra 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 matrix and scalar field, characteristic polynomial and eigenvalues, algebraic and geometric multiplicities, eigenspace dimensions, missing eigenvector count, diagonalizability test, generalized eigenvectors and Jordan blocks and numerical sensitivity are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of linear algebra because they reuse the typed linear algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, For at least one eigenvalue, geometric multiplicity is smaller than algebraic multiplicity, so eigenvectors cannot span the space and a nontrivial Jordan chain is required., and type the carrier, state every parameter and convention in the definition, test that the square matrix and scalar field, characteristic polynomial and eigenvalues, algebraic and geometric multiplicities, eigenspace dimensions, missing eigenvector count, diagonalizability test, generalized eigenvectors and Jordan blocks and numerical sensitivity are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Defective matrix Domain-specific
Parents (1) — more general patterns this builds on
-
Defective matrix is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- Defective matrix → Constraint
Neighborhood in Abstraction Space¶
Defective matrix sits in a crowded region of the domain-specific corpus (3rd 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
- Z-matrix (mathematics) — 0.95
- M-matrix — 0.94
- Matrix congruence — 0.94
- Linear complex structure — 0.94
- Modal matrix — 0.94
Computed from structural-signature embeddings · 2026-09-08