Jacobi's theorem on determinants
E702054
Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in linear algebra ⓘ |
| appearsIn |
advanced linear algebra textbooks
ⓘ
treatises on determinant theory ⓘ |
| appliesTo | square matrices ⓘ |
| assumes | matrix is invertible ⓘ |
| field |
determinant theory
ⓘ
linear algebra ⓘ matrix theory ⓘ |
| formalizes | relationship between determinants of complementary submatrices ⓘ |
| gives | identities between minors of a matrix and minors of its adjugate ⓘ |
| hasConcept |
complementary principal minor
ⓘ
index sets of rows and columns ⓘ principal minor ⓘ submatrix ⓘ |
| holdsFor |
complex matrices
ⓘ
matrices over a commutative field ⓘ real matrices ⓘ |
| implies | relations between principal minors and complementary principal minors ⓘ |
| involves |
adjugate matrix
ⓘ
cofactor matrix ⓘ complementary minors ⓘ determinant ⓘ matrix minors ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi NERFINISHED ⓘ |
| namedEntityType | theorem ⓘ |
| relatedTo |
Cramer's rule
NERFINISHED
ⓘ
Laplace expansion of determinants NERFINISHED ⓘ adjugate-inverse identity A·adj(A)=det(A)I ⓘ cofactor expansion ⓘ |
| relates |
minors of a matrix
ⓘ
minors of the adjugate matrix ⓘ minors of the inverse matrix ⓘ |
| timePeriod | 19th century mathematics ⓘ |
| usedBy |
engineers working with matrix methods
ⓘ
mathematicians ⓘ theoretical physicists ⓘ |
| usedFor |
computations involving minors
ⓘ
deriving determinant identities ⓘ studying properties of the adjugate matrix ⓘ studying properties of the inverse matrix ⓘ |
| usedIn |
classical invariant theory
ⓘ
multilinear algebra ⓘ theory of linear systems ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Carl Gustav Jacob Jacobi