Elementary Arithmetic

A + B: MtrxSprs, MtrxSprs -> MtrxSprs

Given \(m \times n\) sparse matrices \(A\) and \(B\) over a ring \(R\), return \(A+B\).

A - B: MtrxSprs, MtrxSprs -> MtrxSprs

Given \(m \times n\) sparse matrices \(A\) and \(B\) over a ring \(R\), return \(A-B\).

A * B: MtrxSprs, MtrxSprs -> MtrxSprs

Given an \(m \times n\) sparse matrix \(A\) over a ring \(R\) and an \(n \times p\) sparse matrix \(B\) over \(R\), return the \(m \times p\) sparse matrix \(A\cdot B\) over \(R\).

x * A: RngElt, MtrxSprs -> MtrxSprs

A * x: MtrxSprs, RngElt -> MtrxSprs

Given an \(m \times n\) sparse matrix \(A\) over a ring \(R\) and a ring element \(x\) coercible into \(R\), return the scalar product \(x\cdot A\).

-A: MtrxSprs -> MtrxSprs

Given a sparse matrix \(A\), return \(-A\).

A ^-1: MtrxSprs, RngIntElt -> MtrxSprs

Given a invertible square sparse matrix \(A\) over a ring \(R\), return the inverse \(B\) of \(A\) so that \(A\cdot B = B\cdot A = 1\). The coefficient ring \(R\) must be either a field, a Euclidean domain, or a ring with an exact division algorithm and having characteristic equal to zero or greater than \(m\) (this includes most commutative rings).

A ^ n: MtrxSprs, RngIntElt -> MtrxSprs

Given a square sparse matrix \(A\) over a ring \(R\) and an integer \(n\), return the matrix power \(A^n\). \(A^0\) is defined to be the identity matrix for any square matrix \(A\) (even if \(A\) is zero). If \(n\) is negative, \(A\) must be invertible (see the previous function), and the result is \((A^{-1})^{-n}\).

Transpose(A): MtrxSprs -> MtrxSprs

Given an \(m \times n\) sparse matrix \(A\) over a ring \(R\), return the transpose of \(A\), which is simply the \(n \times m\) sparse matrix over \(R\) whose \((i,j)\)-th entry is the \((j,i)\)-th entry of \(A\).