24.9 Echelonized Matrices

A matrix over a field F is in semi-echelon form if the first nonzero element in each row is the identity of F, and all values exactly below these pivots are the zero of F.

SemiEchelonMat returns a record that contains information about a semi-echelonized form of the matrix mat.

The components of this record are

vectors

list of row vectors, each with pivot element the identity of F,

heads

list that contains at position i, if nonzero, the number of the row for that the pivot element is in column i.

This does the same as SemiEchelonMat( mat ), except that it may (and probably will) destroy the matrix mat.

gap> mm:=[[1,2,3],[4,5,6],[7,8,9]];;
gap> SemiEchelonMatDestructive( mm );
rec( heads := [ 1, 2, 0 ], vectors := [ [ 1, 2, 3 ], [ 0, 1, 2 ] ] )
gap> mm;
[ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ]

does the same as SemiEchelonMat but additionally stores the linear transformation T performed on the matrix. The additional components of the result are

coeffs

a list of coefficients vectors of the vectors component, with respect to the rows of mat, that is, coeffs * mat is the vectors component.

relations

a list of basis vectors for the (left) null space of mat.

gap> SemiEchelonMatTransformation([[1,2,3],[0,0,1]]);
rec(
  heads := [ 1, 0, 2 ],
  vectors := [ [ 1, 2, 3 ], [ 0, 0, 1 ] ],
  coeffs := [ [ 1, 0 ], [ 0, 1 ] ],
  relations := [  ] )

A list of matrices over a field F is in semi-echelon form if the list of row vectors obtained on concatenating the rows of each matrix is a semi-echelonized matrix (see SemiEchelonMat).

SemiEchelonMats returns a record that contains information about a semi-echelonized form of the list mats of matrices.

The components of this record are

vectors

list of matrices, each with pivot element the identity of F,

heads

matrix that contains at position [i,j], if nonzero, the number of the matrix that has the pivot element in this position

Does the same as SemiEchelonmats, except that it may estroy its argument. Therefore the argument must be a list of matrices that re mutable.

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