# Accessing or Modifying Entries ## Indexing The following functions and operators enable one to access individual entries or rows of matrices or vectors. ### `A[i]: Mtrx, RngIntElt -> ModTupRngElt` Given a matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, and an integer $i$ such that $1 \leq i \leq m$, return the $i$-th row of $A$, as a vector of length $n$. ### `A[i, j]: Mtrx, RngIntElt, RngIntElt -> RngElt` Given a matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, integers $i$ and $j$ such that $1 \leq i \leq m$ and $1 \leq j \leq n$, return the $(i,j)$-th entry of $A$, as an element of the ring $R$. ### `A[Q]: Mtrx, [ RngIntElt ] -> RngElt` ### `A[i .. j]: Mtrx, RngIntElt, RngIntElt -> RngElt` Given a matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, and a sequence $Q$ of integers in the range $[1..m]$, return the sequence consisting of the rows of $A$ specified by $Q$. This is equivalent to `[A[i]: i in Q]]`. If $Q$ is a range, then the second form `A[i .. j]` may be used to specify the range directly. ### `A[i] := v: Mtrx, RngIntElt, Mtrx` Given a matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, an integer $i$ such that $1 \leq i \leq m$, and a vector $v$ over $R$ of length $n$, modify the $i$-th row of $A$ to be $v$. The integer $0$ may also be given for $v$, indicating the zero vector. ### `A[i, j] := x: Mtrx, RngIntElt, RngIntElt, RngElt` Given a matrix $A$ over the ring $R$ having $m$ rows and $n$ columns, integers $i$ and $j$ such that $1 \leq i \leq m$ and $1 \leq j \leq n$, and a ring element $x$ coercible into $R$, modify the $(i,j)$-th entry of $A$ to be $x$. ### `Example: Indexing (ex-e9c35f)` This example demonstrates simple ways of accessing the entries of matrices. ```magma > X := Matrix(4, [1,2,3,4, 5,4,3,2, 1,2,3,4]); > X; [1 2 3 4] [5 4 3 2] [1 2 3 4] > X[1]; (1 2 3 4) > X[1, 2]; 2 > X[1, 2] := 23; > X; [ 1 23 3 4] [ 5 4 3 2] [ 1 2 3 4] > X[3] := Vector([9,8,7,6]); > X[2] := 0; > X; [ 1 23 3 4] [ 0 0 0 0] [ 9 8 7 6] ``` ## Extracting and Inserting Blocks The following functions enable the extraction of certain rows, columns or general submatrices, or the replacement of a block by another matrix. ### `Submatrix(A, i, j, p, q): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx` ### `ExtractBlock(A, i, j, p, q): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$, $j$, $p$ and $q$ such that $1\leq i \leq i + p \leq m+1$ and $1 \leq j \leq j + q \leq n+1$, return the $p \times q$ submatrix of $A$ rooted at $(i, j)$. Either or both of $p$ and $q$ may be zero, while $i$ may be $m+1$ if $p$ is zero and $j$ may be $n+1$ if $q$ is zero. ### `SubmatrixRange(A, i, j, r, s): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx` ### `ExtractBlockRange(A, i, j, r, s): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$, $j$, $r$ and $s$ such that $1\leq i$, $i-1 \leq r \leq m$, $1 \leq j$, and $j-1 \leq s \leq n$, return the $r-i+1\times s-j+1$ submatrix of $A$ rooted at the $(i, j)$-th entry and extending to the $(r, s)$-th entry, inclusive. $r$ may equal $i-1$ or $s$ may equal $j-1$, in which case a matrix with zero rows or zero columns, respectively, will be returned. ### `Submatrix(A, I, J): Mtrx, [RngIntElt], [RngIntElt] -> Mtrx` Given an $m \times n$ matrix $A$ and integer sequences $I$ and $J$, return the submatrix of $A$ given by the row indices in $I$ and the column indices in $J$. ### `InsertBlock(A, B, i, j): Mtrx, Mtrx, RngIntElt, RngIntElt -> Mtrx` ### `InsertBlock(~A, B, i, j): Mtrx, Mtrx, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ over a ring $R$, a $p \times q$ matrix $B$ over $R$, and integers $i$ and $j$ such that $1\leq i \leq i + p \leq m+1$ and $1 \leq j \leq j + q \leq n+1$, insert $B$ at position $(i,j)$ in $A$. In the functional version ($A$ is a value argument), this function returns the new matrix and leaves $A$ untouched, while in the procedural version ($\sim A$ is a reference argument), $A$ is modified in place so that the $p \times q$ submatrix of $A$ rooted at $(i, j)$ is now equal to $B$. ### `RowSubmatrix(A, i, k): Mtrx, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$ and $k$ such that $1 \le i \leq i + k \leq m+1$, return the $k \times n$ submatrix of $X$ consisting of rows $[i \ldots i + k - 1]$ inclusive. The integer $k$ may be zero and $i$ may also be $m+1$ if $k$ is zero, but the result will always have $n$ columns. ### `RowSubmatrix(A, i): Mtrx, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and an integer $i$ such that $0 \le i \leq m$, return the $i\times n$ submatrix of $X$ consisting of the first $i$ rows. The integer $i$ may be $0$, but the result will always have $n$ columns. ### `RowSubmatrixRange(A, i, j): Mtrx, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$ and $j$ such that $1 \le i$ and $i-1 \leq j \leq m$, return the $j-i+1 \times n$ submatrix of $X$ consisting of rows $[i \ldots j]$ inclusive. The integer $j$ may equal $i-1$, in which case a matrix with zero rows and $n$ columns will be returned. ### `ColumnSubmatrix(A, i, k): Mtrx, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$ and $k$ such that $1 \le i \leq i + k \leq n+1$, return the $m \times k$ submatrix of $X$ consisting of columns $[i \ldots i + k - 1]$ inclusive. The integer $k$ may be zero and $i$ may also be $n+1$ if $k$ is zero, but the result will always have $m$ rows. ### `ColumnSubmatrix(A, i): Mtrx, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and an integer $i$ such that $0 \le i \leq n$, return the $m\times i$ submatrix of $X$ consisting of the first $i$ columns. The integer $i$ may be $0$, but the result will always have $m$ rows. ### `ColumnSubmatrixRange(A, i, j): Mtrx, RngIntElt, RngIntElt -> Mtrx` Given an $m \times n$ matrix $A$ and integers $i$ and $j$ such that $1 \le i$ and $i-1 \leq j \leq n$, return the $m\times j-i+1$ submatrix of $X$ consisting of columns $[i \ldots j]$ inclusive. The integer $j$ may equal $i-1$, in which case a matrix with zero columns and $n$ rows will be returned. ### `Example: Submatrix (ex-0ed08c)` The use of the submatrix operations is illustrated by applying them to a $6 \times 6$ matrix over the ring of integers ${\mathbb{Z}}$. ```magma > A := Matrix(6, > [ 9, 1, 7, -3, 2, -1, > 3, -4, -5, 9, 2, 7, > 7, 1, 0, 1, 8, 22, > -3, 3, 3, 8, 8, 37, > -9, 0, 7, -1, 2, 3, > 7, 2, -2, 4, 3, 47 ]); > A; [ 9 1 7 -3 2 -1] [ 3 -4 -5 9 2 7] [ 7 1 0 1 8 22] [-3 3 3 8 8 37] [-9 0 7 -1 2 3] [ 7 2 -2 4 3 47] > Submatrix(A, 2,2, 3,3); [-4 -5 9] [ 1 0 1] [ 3 3 8] > SubmatrixRange(A, 2,2, 3,3); [-4 -5] [ 1 0] > S := $1; > InsertBlock(~A, S, 5,5); > A; [ 9 1 7 -3 2 -1] [ 3 -4 -5 9 2 7] [ 7 1 0 1 8 22] [-3 3 3 8 8 37] [-9 0 7 -1 -4 -5] [ 7 2 -2 4 1 0] > RowSubmatrix(A, 5, 2); [-9 0 7 -1 -4 -5] [ 7 2 -2 4 1 0] > RowSubmatrixRange(A, 2, 3); [ 3 -4 -5 9 2 7] [ 7 1 0 1 8 22] > RowSubmatrix(A, 2, 0); Matrix with 0 rows and 6 columns ``` ## Row and Column Operations The following functions and procedures provide elementary row or column operations on matrices. For each operation, there is a corresponding function which creates a new matrix for the result (leaving the input matrix unchanged), and a corresponding procedure which modifies the input matrix in place. ### `SwapRows(A, i, j): Mtrx, RngIntElt, RngIntElt -> Mtrx` ### `SwapRows(~A, i, j): Mtrx, RngIntElt, RngIntElt` Given an $m \times n$ matrix $A$ and integers $i$ and $j$ such that $1 \le i\le m$ and $1 \leq j \leq m$, swap the $i$-th and $j$-th rows of $A$. ### `SwapColumns(A, i, j): Mtrx, RngIntElt, RngIntElt -> Mtrx` ### `SwapColumns(~A, i, j): Mtrx, RngIntElt, RngIntElt` Given an $m \times n$ matrix $A$ and integers $i$ and $j$ such that $1 \le i\le n$ and $1 \leq j \leq n$, swap the $i$-th and $j$-th columns of $A$. ### `ReverseRows(A): Mtrx -> Mtrx` ### `ReverseRows(~A): Mtrx` Given a matrix $A$, reverse all the rows of $A$. ### `ReverseColumns(A): Mtrx -> Mtrx` ### `ReverseColumns(~A): Mtrx` Given a matrix $A$, reverse all the columns of $A$. ### `AddRow(A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt -> Mtrx` ### `AddRow(~A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt` Given an $m \times n$ matrix $A$ over a ring $R$, a ring element $c$ coercible into $R$, and integers $i$ and $j$ such that $1 \le i\le m$ and $1 \leq j \leq m$, add $c$ times row $i$ of $A$ to row $j$ of $A$. ### `AddColumn(A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt -> Mtrx` ### `AddColumn(~A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt` Given an $m \times n$ matrix $A$ over a ring $R$, a ring element $c$ coercible into $R$, and integers $i$ and $j$ such that $1 \le i\le n$ and $1 \leq j \leq n$, add $c$ times column $i$ of $A$ to column $j$. ### `MultiplyRow(A, c, i): Mtrx, RngElt, RngIntElt -> Mtrx` ### `MultiplyRow(~A, c, i): Mtrx, RngElt, RngIntElt` Given an $m \times n$ matrix $A$ over a ring $R$, a ring element $c$ coercible into $R$, and an integer $i$ such that $1 \le i\le m$, multiply row $i$ of $A$ by $c$ (on the left). ### `MultiplyColumn(A, c, i): Mtrx, RngElt, RngIntElt -> Mtrx` ### `MultiplyColumn(~A, c, i): Mtrx, RngElt, RngIntElt` Given an $m \times n$ matrix $A$ over a ring $R$, a ring element $c$ coercible into $R$, and an integer $i$ such that $1 \le i\le n$, multiply column $i$ of $A$ by $c$ (on the left). ### `RemoveRow(A, i): Mtrx, RngIntElt -> Mtrx` ### `RemoveRow(~A, i): Mtrx, RngIntElt` Given an $m \times n$ matrix $A$ and an integer $i$ such that $1 \le i\le m$, remove row $i$ from $A$ (leaving an $(m-1) \times n$ matrix). ### `RemoveColumn(A, j): Mtrx, RngIntElt -> Mtrx` ### `RemoveColumn(~A, j): Mtrx, RngIntElt` Given an $m \times n$ matrix $A$ and an integer $j$ such that $1 \le j\le n$, remove column $j$ from $A$ (leaving an $m \times (n-1)$ matrix). ### `RemoveRowColumn(A, i, j): Mtrx, RngIntElt -> Mtrx` ### `RemoveRowColumn(~A, i, j): Mtrx, RngIntElt` Given an $m \times n$ matrix $A$ and integers $i$ and $j$ such that $1 \le i\le m$ and $1 \le j\le n$, remove row $i$ and column $j$ from $A$ (leaving an $(m-1) \times (n-1)$ matrix). ### `RemoveZeroRows(A): Mtrx -> Mtrx` ### `RemoveZeroRows(~A): Mtrx` Given a matrix $A$, remove all the zero rows of $A$. ### `Example: Row Column Ops (ex-80d37c)` The use of row and column operations is illustrated by applying them to a $5 \times 6$ matrix over the ring of integers ${\mathbb{Z}}$. ```magma > A := Matrix(5, 6, > [ 3, 1, 0, -4, 2, -12, > 2, -4, -5, 5, 23, 6, > 8, 0, 0, 1, 5, 12, > -2, -6, 3, 8, 9, 17, > 11, 12, -6, 4, 2, 27 ]); > A; [ 3 1 0 -4 2 -12] [ 2 -4 -5 5 23 6] [ 8 0 0 1 5 12] [ -2 -6 3 8 9 17] [ 11 12 -6 4 2 27] > SwapColumns(~A, 1, 2); > A; [ 1 3 0 -4 2 -12] [ -4 2 -5 5 23 6] [ 0 8 0 1 5 12] [ -6 -2 3 8 9 17] [ 12 11 -6 4 2 27] > AddRow(~A, 4, 1, 2); > AddRow(~A, 6, 1, 4); > AddRow(~A, -12, 1, 5); > A; [ 1 3 0 -4 2 -12] [ 0 14 -5 -11 31 -42] [ 0 8 0 1 5 12] [ 0 16 3 -16 21 -55] [ 0 -25 -6 52 -22 171] > RemoveRow(~A, 1); > A; [ 2 -4 -5 5 23 6] [ 8 0 0 1 5 12] [-2 -6 3 8 9 17] [11 12 -6 4 2 27] > RemoveRowColumn(~A, 4, 6); > A; [ 2 -4 -5 5 23] [ 8 0 0 1 5] [-2 -6 3 8 9] ```