Gaussian elimination
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linalg::gaussElim(A
, <All>)
linalg::gaussElim(A)
performs Gaussian
elimination on the matrix A to
reduce A to
a similar matrix in upper row echelon form.
A row echelon form of A
returned by linalg::gaussElim
is
not unique. See linalg::gaussJordan
for computing the reduced row
echelon form.
The component ring R of A
must
be an integral domain, i.e., a domain of category Cat::IntegralDomain
.
If R is
a field, i.e., a domain of category Cat::Field
, ordinary Gaussian
elimination is used. Otherwise, linalg::gaussElim
applies
fractionfree Gaussian elimination to A
.
linalg::gaussElim
serves as an interface
function for the method "gaussElim"
of the matrix
domain of A
, i.e., one may call A::dom::gaussElim(A)
directly
instead of linalg::gaussElim(A, All)
Refer to the help page of Dom::Matrix
for details
about the computation strategy of linalg::gaussElim
.
We apply Gaussian elimination to the following matrix:
A := Dom::Matrix(Dom::Rational)( [[1, 2, 3, 4], [1, 0, 1, 0], [3, 5, 6, 9]] )
which reduces A
to the following row echelon
form:
linalg::gaussElim(A)
We apply Gaussian elimination to the matrix:
B := Dom::Matrix(Dom::Integer)( [[1, 2, 1], [1, 0, 1], [2, 1, 4]] )
and get the following result:
linalg::gaussElim(B, All)
We see that rank(B) = 3 and .

A matrix of a domain of category 

Returns a list
where T is
a row echelon form of If 
a matrix of the same domain type as A
, or
the list [T, rank(A), det(A), {j_1,dots,j_r}]
when
the option All
is given (see below).
Let T = (t_{i, j})_{1 ≤ i ≤ m, 1 ≤ j ≤ n} be an m×n matrix. Then T is a matrix in an upper row echelon form, if r ∈ {0, 1, …, n} and indices j_{1}, j_{2}, …, j_{r} ∈ {1, …, n} exist with:
j_{1} < j_{2} < ··· < j_{r}.
For each i ∈ {1, …, r}: t_{i, 1} = t_{i, 2} = ··· = t_{i, ji  1} = 0.
For each i ∈ {r + 1, …, m}: t_{i, j} = 0 for each j ∈ {1, …, n}.
The indices j_{1}, j_{2}, …, j_{r} are the characteristic column indices of the matrix T.