Matrix accessors

• M[i,j] read the (i,j)-entry of matrix M
• SetEntry(M,i,j, val) set the (i,j)-entry of matrix M
• GetRow(M,i) return the i-th row of M as a vector<RingElem>
• GetCol(M,j) return the j-th col of M as a vector<RingElem>
• GetRows(M) return the rows of M as a vector<vector<RingElem>>
• GetCols(M) return the cols of M as a vector<vector<RingElem>>
• FlattenByRows(M) return entries of M in a vector<RingElem> in order 1st row, 2nd row, etc
• FlattenByCols(M) return entries of M in a vector<RingElem> in order 1st col, 2nd col, etc
  Note that GetRow, GetCol, GetRows, GetCols, FlattenByRows and FlattenByCols make copies of the matrix entries.

Matrix Arithmetic

There are two ways of multiplying two matrices together. The infix operators return a DenseMatrix; the procedural version may be slightly faster than the infix operator.

• mul(matrix& lhs, M1, M2) -- a procedure equivalent to lhs = M1*M2;, note that lhs might be a SparseMatrix (not yet implemented)
• operator*(M1, M2) -- the product M1*M2
• operator+(M1, M2) -- the sum M1+M2
• operator-(M1, M2) -- the difference M1-M2
• power(M, n) compute n-th power of M; if n is negative then M must be invertible
• operator*(n, M1) -- scalar multiple of M1 by n (integer or RingElem)
• operator*(M1, n) -- scalar multiple of M1 by n (integer or RingElem)
• operator/(M1, n) -- scalar multiple of M1 by 1/n (where n is integer or RingElem)
• operator-(M1) -- scalar multiple of M1 by -1

Matrix norms

Here are some matrix norms. The result is an element of the ring containing the matrix elements. Note that FrobeniusNormSq gives the square of the Frobenius norm (so that the value surely lies in the same ring).

• FrobeniusNormSq(M) -- the square of the Frobenius norm
• OperatorNormInfinity(M) -- the infinity norm, ring must be ordered
• OperatorNorm1(M) -- the one norm, ring must be ordered

Sundry functions

Here are some fairly standard functions on matrices.

• det(M) -- determinant of M (M must be square)
• IsZeroDet(M) -- equivalent to IsZero(det(M)) (but may be faster)
• HadamardBoundSq(M) -- computes row and column bounds in a struct (fields myRowBound and myColBound)
• rk(M) -- rank of M (the base ring must be an integral domain)
• inverse(M) -- inverse of M as a DenseMatrix
• adj(M) -- classical adjoint of M as a DenseMatrix; sometimes called "adjugate".
• rref(M) -- compute a reduced row echelon form of M (orig. matrix is unchanged); matrix must be over a field
• PseudoInverse(M) -- PseudoInverse of M as a DenseMatrix. I suspect that it requires that the matrix be of full rank.
• LinSolve(M,rhs) -- solve for x the linear system M*x = rhs; result is a DenseMatrix; if no soln exists, result is the 0-by-0 matrix
• LinKer(M) -- solve for x the linear system M*x = 0; returns a DenseMatrix whose columns are a base for ker(M)
• LinKerZZ(M) -- solve for x the linear system M*x = 0; returns a DenseMatrix whose columns are a ZZ-base for integer points in ker(M)