Predicates

The functions in this section test various properties of matrices. See also the Lattices chapter for a description of the function IsPositiveDefinite and related functions.

IsZero(A): Mtrx -> BoolElt

Given an \(m \times n\) matrix \(A\) over the ring \(R\), return true iff \(A\) is the \(m \times n\) zero matrix.

IsOne(A): Mtrx -> BoolElt

Given a square \(m \times m\) matrix \(A\) over the ring \(R\), return true iff \(A\) is the \(m \times m\) identity matrix.

IsMinusOne(A): Mtrx -> BoolElt

Given a square \(m \times m\) matrix \(A\) over the ring \(R\), return true iff \(A\) is the negation of the \(m \times m\) identity matrix.

IsScalar(A): Mtrx -> BoolElt

Given a square \(m \times m\) matrix \(A\) over the ring \(R\), return true iff \(A\) is scalar, i.e., iff \(A\) is the product of some element of \(R\) and the \(m \times m\) identity matrix.

IsDiagonal(A): Mtrx -> BoolElt

Given a square matrix \(A\) over the ring \(R\), return true iff \(A\) is diagonal, i.e., iff the only non-zero entries of \(A\) are on the diagonal.

IsSymmetric(A): Mtrx -> BoolElt

Given a square matrix \(A\) over the ring \(R\), return true iff \(A\) is symmetric, i.e., iff \(A\) equals its transpose.

IsHermitian(A, f): Mtrx, Map -> BoolElt

Given a square matrix \(A\) over the ring \(R\), and an involution \(f : R \to R\), return true iff \(A\) is Hermitian with respect to \(f\), i.e., iff \(A\) equals applying \(f\) to its transpose.

IsUpperTriangular(A): Mtrx -> BoolElt

Given a matrix \(A\) over the ring \(R\), return true iff \(A\) is upper triangular, i.e., iff the only non-zero entries of \(A\) are on or above the diagonal.

IsLowerTriangular(A): Mtrx -> BoolElt

Given a matrix \(A\) over the ring \(R\), return true iff \(A\) is lower triangular, i.e., iff the only non-zero entries of \(A\) are on or below the diagonal.

IsUnit(A): Mtrx -> BoolElt

Given a square matrix \(A\) over the ring \(R\), return true iff \(A\) is a unit, i.e., iff \(A\) has an inverse. The coefficient ring \(R\) may be any commutative ring (since the computation depends on testing if the determinant is a unit – a calculation which is supported in all commutative rings).

IsSingular(A): Mtrx -> BoolElt

Given a square \(m \times m\) matrix \(A\) over the ring \(R\), return true iff \(A\) is singular, i.e., iff the determinant of \(A\) is zero (or, equivalently, iff the rank of \(A\) is less than \(m\)). Note that (not IsSingular(A)) is not equivalent to IsUnit(A) whenever \(R\) is not a field: if the determinant of \(A\) is non-zero but not a unit, then \(A\) is non-singular but not invertible. The coefficient ring \(R\) may be any commutative ring (since the computation involves only computing the determinant and testing whether it is zero).

IsSymplecticMatrix(A): Mtrx -> BoolElt

Given an \(m \times m\) matrix \(A\) over the integers, return true if and only if \(A\) is an integer symplectic matrix, that is, \(AJ{}^tA = J\), where \(J = \left(\begin{matrix}0 & {\bf 1}_g \\ -{\bf 1}_g & 0\end{matrix}\right).\)