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Gausselim

inert Gaussian elimination

Gaussjord

inert Gauss Jordan elimination

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Gausselim(A) mod p

Gaussjord(A) mod p

Gausselim(A, 'r', 'd') mod p

Gaussjord(A, 'r', 'd') mod p

Parameters

A

-

Matrix

'r'

-

(optional) for returning the rank of A

'd'

-

(optional) for returning the determinant of A

'p'

-

an integer, the modulus

Description

• 

The Gausselim and Gaussjord functions are placeholders for representing row echelon forms of the rectangular matrix A.

• 

The commands Gausselim(A,...) mod p and Gassjord(A,...) mod p apply Gaussian elimination with row pivoting to A, a rectangular matrix over a finite ring of characteristic p. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOfs.

• 

The result of the Gausselim command is a an upper triangular matrix B in row echelon form.  The result of the Gaussjord command is also an upper triangular matrix B but in reduced row echelon form.

• 

If an optional second parameter is specified, and it is a name, it is assigned the rank of the matrix A.

• 

If A is an m by n matrix with mn and if an optional third parameter is also specified, and it is a name, it is assigned the determinant of the matrix A[1..m,1..m].

Examples

A := Matrix([[1,2,3],[1,3,0],[1,4,3]]);

A123130143

(1)

Gausselim(A) mod 5;

123012001

(2)

B := ArrayTools[Concatenate](2,A,LinearAlgebra[IdentityMatrix](3));

B123100130010143001

(3)

Gaussjord(B) mod 5;

100411010203001131

(4)

Inverse(A) mod 5;

411203131

(5)

alias(a=RootOf(x^4+x+1) mod 2): # GF(2^4)

A := Matrix([[1,a,a^2],[a,a^2,a^3],[a^2,a^3,1]]);

A1aa2aa2a3a2a31

(6)

Gausselim(A,'r','d') mod 2;

1aa200a000

(7)

r;

2

(8)

d;

0

(9)

See Also

Det

Inverse

LinearAlgebra[GaussianElimination]

LinearAlgebra[Modular]

mod

Modular[RowReduce]