Gaussian elimination
E29360
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
Observed surface forms (3)
| Surface form | Occurrences |
|---|---|
| Gauss–Jordan elimination | 2 |
| Gaussian elimination with full pivoting | 1 |
| Gaussian elimination with partial pivoting | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm
ⓘ
linear algebra method ⓘ matrix algorithm ⓘ |
| applicableTo |
systems with infinitely many solutions
ⓘ
systems with no solution ⓘ systems with unique solutions ⓘ |
| assumes | arithmetic operations are exact in theoretical analysis ⓘ |
| basedOn | elementary row operations ⓘ |
| canDetect |
linear dependence of rows
ⓘ
singularity of a matrix ⓘ |
| field | linear algebra ⓘ |
| historicalOrigin | methods known in ancient Chinese mathematics ⓘ |
| input |
coefficient matrix of a linear system
ⓘ
right-hand side vector of a linear system ⓘ |
| limitation | can be numerically unstable without pivoting ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| numericalVariant |
Gaussian elimination
self-linksurface differs
ⓘ
surface form:
Gaussian elimination with full pivoting
Gaussian elimination self-linksurface differs ⓘ
surface form:
Gaussian elimination with partial pivoting
|
| operatesOn |
augmented matrices
ⓘ
matrices ⓘ |
| output |
row echelon form of a matrix
ⓘ
solution of a linear system ⓘ |
| property | preserves solution set of the linear system ⓘ |
| relatedTo |
Gaussian elimination
self-linksurface differs
ⓘ
surface form:
Gauss–Jordan elimination
LU decomposition ⓘ reduced row echelon form ⓘ row echelon form ⓘ |
| step |
back substitution
ⓘ
forward elimination ⓘ |
| timeComplexity | O(n^3) for an n by n system ⓘ |
| usedFor |
computing determinants
ⓘ
computing matrix rank ⓘ finding inverses of matrices ⓘ reducing matrices to reduced row echelon form ⓘ reducing matrices to row echelon form ⓘ solving systems of linear equations ⓘ |
| usedIn |
computer graphics
ⓘ
data analysis ⓘ engineering ⓘ numerical linear algebra libraries ⓘ scientific computing ⓘ |
| usesOperation |
row replacement
ⓘ
row scaling ⓘ row swapping ⓘ |
| worksOver |
complex numbers
ⓘ
fields ⓘ finite fields ⓘ real numbers ⓘ |
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gauss–Jordan elimination
this entity surface form:
Gaussian elimination with partial pivoting
this entity surface form:
Gaussian elimination with full pivoting
this entity surface form:
Gauss–Jordan elimination