# 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 $A$.