Minor (linear algebra)¶
The determinant of a square submatrix obtained by selecting equal-size subsets of a matrix’s rows and columns.
Core Idea¶
A k-order minor is indexed by k rows and k columns and equals the determinant of their intersection submatrix; complementary and first minors specialize the selection. Row-column selection exposes local determinant structure used in rank tests, cofactors, inverse formulas, exterior powers, and determinantal varieties. 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.
The load-bearing residual is not the broad topic of linear algebra. It is the domain-specific identity determined by the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix.
Scope of Application¶
Minor (linear algebra) belongs to linear algebra and is useful where the analyst can specify the typed linear algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, then evaluate the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix. The scope is broad within that domain but bounded by the need for the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Minor (linear algebra) can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
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 Minor (linear algebra). Minor (linear algebra) 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, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of linear algebra because they reuse the typed linear algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, Row-column selection exposes local determinant structure used in rank tests, cofactors, inverse formulas, exterior powers, and determinantal varieties., and type the carrier, state every parameter and convention in the definition, test that the selected row and column index sets have equal cardinality and the reported scalar is exactly the determinant of the induced square submatrix, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Minor (linear algebra) Domain-specific
Parents (1) — more general patterns this builds on
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Minor (linear algebra) is a kind of Decomposition Prime
The proposed strict upward parent is
prime:decomposition.
Hierarchy path (1) — routes to 1 parentless root
- Minor (linear algebra) → Decomposition
Neighborhood in Abstraction Space¶
Minor (linear algebra) sits in a crowded region of the domain-specific corpus (27th 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
- Matrix congruence — 0.91
- Unimodular matrix — 0.91
- Exchange matrix — 0.91
- Cauchy matrix — 0.90
- Crout matrix decomposition — 0.90
Computed from structural-signature embeddings · 2026-09-08