# Accessing Sparse Matrices

The following functions access basic properties of sparse matrices.

## Elementary Properties

### `BaseRing(A): MtrxSprs -> Rng`

### `CoefficientRing(A): MtrxSprs -> Rng`

Given a sparse matrix $A$ with entries lying in a ring $R$, return $R$.

### `NumberOfRows(A): MtrxSprs -> RngIntElt`

### `Nrows(A): MtrxSprs -> RngIntElt`

Given an $m \times n$ sparse matrix $A$, return $m$, the number of rows of $A$.

### `NumberOfColumns(A): MtrxSprs -> RngIntElt`

### `Ncols(A): MtrxSprs -> RngIntElt`

Given an $m \times n$ sparse matrix $A$, return $n$, the number of columns of $A$.

### `ElementToSequence(A): MtrxSprs -> [ <RngIntElt, RngIntElt, RngElt> ]`

### `Eltseq(A): MtrxSprs -> [ <RngIntElt, RngIntElt, RngElt> ]`

Given a sparse matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, return the entries of $A$ as a sequence of all tuples of the form `<i, j, x>` such that the $[i, j]$-th entry of $A$ equals $x$ and $x$ is non-zero. It is always true that `SparseMatrix(Nrows(A), Ncols(A), Eltseq(A))` equals $A$.

### `NumberOfNonZeroEntries(A): MtrxSprs -> RngIntElt`

### `NNZEntries(A): MtrxSprs -> RngIntElt`

Given a sparse matrix $A$, return the number of non-zero entries in $A$.

### `Density(A): MtrxSprs -> FldRe`

Given a sparse matrix $A$, return the density of $A$ as a real number, which is the number of non-zero entries in $A$ divided by the product of the number of rows of $A$ and the number of columns of $A$ (or zero if $A$ has zero rows or columns).

### `Support(A, i): MtrxSprs, RngIntElt -> [RngIntElt]`

Given a sparse matrix $A$ having $r$ rows, and an integer $i$ such that $1\leq i\leq r$, return the support of row $i$ of $A$; i.e., the column numbers of the non-zero entries of row $i$ of $A$.

### `Support(A): MtrxSprs -> [ <RngIntElt, RngIntElt, RngElt> ]`

Given a sparse matrix $A$, return the sequence of all pairs `<i, j>` such that the $[i, j]$-th entry of $A$ is non-zero.

## Weights

### `RowWeight(A, i): MtrxSprs, RngIntElt -> RngIntElt`

Given a sparse matrix $A$ with $m$ rows and an integer $i$ such that $1\leq i\leq m$, return the weight (number of non-zero entries) of the $i$-th row of $A$.

### `RowWeights(A): MtrxSprs -> [RngIntElt]`

Given a sparse matrix $A$ with $m$ rows, return the length $m$ sequence of integers whose $i$-th entry is the weight of the $i$-th row of $A$.

### `ColumnWeight(A, j): MtrxSprs, RngIntElt -> RngIntElt`

Given a sparse matrix $A$ with $n$ columns and an integer $j$ such that $1\leq j\leq n$, return the weight (number of non-zero entries) of the $j$-th column of $A$.

### `ColumnWeights(A): MtrxSprs -> [RngIntElt]`

Given a sparse matrix $A$ with $n$ columns, return the length $n$ sequence of integers whose $j$-th entry is the weight of the $j$-th column of $A$.