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)abbrev package IMATQF InnerMatrixQuotientFieldFunctions
++ Author: Clifton J. Williamson
++ Date Created: 22 November 1989
++ Date Last Updated: 22 November 1989
++ Description:
++ \spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices
++ over an integral domain which involve the quotient field of that integral
++ domain.  The functions rowEchelon and inverse return matrices with
++ entries in the quotient field.

InnerMatrixQuotientFieldFunctions(R,Row,Col,M,QF,Row2,Col2,M2) : SIG == CODE where
  R    : IntegralDomain
  Row  : FiniteLinearAggregate R
  Col  : FiniteLinearAggregate R
  M    : MatrixCategory(R,Row,Col)
  QF   : QuotientFieldCategory R
  Row2 : FiniteLinearAggregate QF
  Col2 : FiniteLinearAggregate QF
  M2   : MatrixCategory(QF,Row2,Col2)

  IMATLIN ==> InnerMatrixLinearAlgebraFunctions(QF,Row2,Col2,M2)
  MATCAT2 ==> MatrixCategoryFunctions2(R,Row,Col,M,QF,Row2,Col2,M2)
  CDEN    ==> InnerCommonDenominator(R,QF,Col,Col2)

  SIG ==> with

      rowEchelon : M -> M2
        ++ \spad{rowEchelon(m)} returns the row echelon form of the matrix m.
        ++ the result will have entries in the quotient field.

      inverse : M -> Union(M2,"failed")
        ++ \spad{inverse(m)} returns the inverse of the matrix m.
        ++ If the matrix is not invertible, "failed" is returned.
        ++ Error: if the matrix is not square.
        ++ Note that the result will have entries in the quotient field.

      if Col2 has shallowlyMutable then

        nullSpace : M -> List Col
          ++ \spad{nullSpace(m)} returns a basis for the null space of the
          ++ matrix m.

  CODE ==> add

    qfMat: M -> M2
    qfMat m == map((r1:R):QF +-> r1::QF,m)$MATCAT2

    rowEchelon m == rowEchelon(qfMat m)$IMATLIN

    inverse m ==
      (inv := inverse(qfMat m)$IMATLIN) case "failed" => "failed"
      inv :: M2

    if Col2 has shallowlyMutable then

      nullSpace m ==
        [clearDenominator(v)$CDEN for v in nullSpace(qfMat m)$IMATLIN]