UpperTriangular
E440648
UpperTriangular is a linear algebra type representing square matrices whose entries below the main diagonal are zero, enabling efficient storage and specialized algorithms.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
linear algebra data structure
ⓘ
matrix type ⓘ |
| belongsTo | category of structured matrices ⓘ |
| canBeStoredAs | compact representation of upper triangle including diagonal ⓘ |
| enables |
efficient storage
ⓘ
specialized algorithms ⓘ |
| hasAdvantage |
allows specialized kernels exploiting zero structure
ⓘ
reduces number of stored elements compared to full matrix ⓘ |
| hasConstraint |
indices below diagonal are implicitly zero
ⓘ
matrix must be square ⓘ |
| hasLowerPart | zero entries ⓘ |
| hasMainDiagonal | possibly nonzero entries ⓘ |
| hasProperty |
all nonzero entries are on or above main diagonal
ⓘ
entries below main diagonal are zero ⓘ |
| hasUpperPart | possibly nonzero entries ⓘ |
| hasUseCase |
block upper triangular systems
ⓘ
efficient repeated solves with same triangular matrix ⓘ |
| isAssumedIn | algorithms that skip operations on known zeros below diagonal ⓘ |
| isCharacterizedBy | a_ij = 0 for i > j ⓘ |
| isCommonIn |
R factor of QR factorization
ⓘ
solutions of linear systems after Gaussian elimination ⓘ upper factor of LU factorization ⓘ |
| isDefinedOver | a field or ring of matrix entries ⓘ |
| isImportantFor |
numerical stability analysis of algorithms
ⓘ
theoretical linear algebra proofs involving triangularization ⓘ |
| isRelatedTo |
Diagonal matrix
ⓘ
LowerTriangular ⓘ |
| isSubtypeOf |
square matrix type
ⓘ
triangular matrix type ⓘ |
| isUsedBy | numerical linear algebra libraries ⓘ |
| isUsedIn |
Cholesky decomposition
ⓘ
LU decomposition ⓘ QR decomposition (R factor) ⓘ factorization algorithms ⓘ |
| mayBe |
non-unit upper triangular (arbitrary diagonal)
ⓘ
unit upper triangular (ones on diagonal) ⓘ |
| optimizes |
memory usage for sparse lower part
ⓘ
time complexity of certain matrix operations ⓘ |
| represents | square matrix ⓘ |
| supportsOperation |
matrix-matrix multiplication
ⓘ
matrix-vector multiplication ⓘ solving linear systems with triangular coefficient matrix ⓘ |
| usedFor |
backward substitution
ⓘ
forward substitution ⓘ solving triangular systems ⓘ |
| usedIn | linear algebra computations ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.