Defective matrix¶

A square matrix lacking a full basis of eigenvectors and therefore not diagonalizable over the stated field.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4075
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Linear Algebra → Mathematics

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.

How would you explain it like I'm…

No faithful explanation at this level. Two of three generators (B, C) mark it na: the only child-level picture, a stretching machine lacking enough 'unturned arrows', misclassifies field-dependent cases like plane rotations as defective and drops the per-eigenvalue geometric-vs-algebraic multiplicity comparison that defines it.

Missing Special Directions

A matrix is a grid of numbers that acts like a machine moving arrows around. Some special arrows don't get turned by the machine; they only get stretched or shrunk by a certain amount. A math recipe tells you, for each stretch amount, the most special arrows pointing in truly different directions that it could have. A matrix is defective when, for at least one stretch amount, fewer of those special arrows show up than the recipe allows. Having a repeated stretch amount is needed for this, but it doesn't always cause it.

Non-Diagonalizable (Defective) Matrix

A square matrix is defective when it does not have enough linearly independent eigenvectors to span the whole space, so it cannot be diagonalized. Precisely, for at least one eigenvalue the geometric multiplicity, the dimension of its eigenspace, is smaller than its algebraic multiplicity, the number of times it is a root of the characteristic polynomial. A repeated eigenvalue is necessary for this but not enough: the identity matrix has a repeated eigenvalue but is not defective. The shear matrix with rows (1, 1) and (0, 1) is defective, with eigenvalue 1 repeated twice but only one eigenvector direction. Generalized eigenvectors fill in the missing directions and give a Jordan basis, but that does not make the matrix diagonalizable. Whether a matrix is defective can also depend on which number field you work over.

A square matrix is defective when, for at least one eigenvalue, its geometric multiplicity (the dimension of its eigenspace) is strictly less than its algebraic multiplicity (its multiplicity as a root of the characteristic polynomial). Then the eigenvectors cannot span the space, so the matrix is not diagonalizable, and a nontrivial Jordan chain is needed. Repeated eigenvalues are necessary but not sufficient; the identity matrix, for example, has one repeated eigenvalue yet is diagonal. Defectiveness also depends on the scalar field over which the matrix is considered. Generalized eigenvectors complete a Jordan basis, giving a Jordan form with at least one block larger than 1×1, but they do not make the matrix diagonalizable. The number of missing eigenvectors is the total gap between algebraic and geometric multiplicities, and defective matrices are numerically sensitive, since small perturbations can change their eigenstructure.

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.
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) — outgoing typed relations

Defective matrix is a kind of Constraint Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

Defective matrix → is a kind of Constraint

Neighborhood in Abstraction Space¶

Defective matrix sits in a crowded region of the domain-specific corpus (2^nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Structured Matrices & Linear Maps (43 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25