# Building Block Matrices

Block matrices can be constructed either by listing the blocks, or by joining together smaller matrices horizontally, vertically or diagonally.

## `BlockMatrix(m, n, blocks): RngIntElt, RngIntElt, [ Mtrx ] -> Mtrx`

The matrix constructed from the given block matrices, which should all have the same dimensions, and should be given as a sequence of $m\cdot n$ block matrices (given in row major order, in other words listed across rows).

## `BlockMatrix(m, n, rows): RngIntElt, RngIntElt, [ [ Mtrx ] ] -> Mtrx`

## `BlockMatrix(rows): [ [ Mtrx ] ] -> Mtrx`

The matrix constructed from the given block matrices, which should all have the same dimensions, and should be given as a sequence of $m$ rows, each containing $n$ block matrices.

## `HorizontalJoin(X, Y): Mtrx, Mtrx -> Mtrx`

Given a matrix $X$ with $r$ rows and $c$ columns, and a matrix $Y$ with $r$ rows and $d$ columns, both over the same coefficient ring $R$, return the matrix over $R$ with $r$ rows and $(c+d)$ columns obtained by joining $X$ and $Y$ horizontally (placing $Y$ to the right of $X$).

## `HorizontalJoin(Q): [ Mtrx ] -> Mtrx`

## `HorizontalJoin(T): < Mtrx > -> Mtrx`

Given a sequence $Q$ or tuple $T$ of matrices, each having the same number of rows and being over the same coefficient ring $R$, return the matrix over $R$ obtained by joining the elements of $Q$ or $T$ horizontally in order.

## `VerticalJoin(X, Y): Mtrx, Mtrx -> Mtrx`

Given a matrix $X$ with $r$ rows and $c$ columns, and a matrix $Y$ with $s$ rows and $c$ columns, both over the same coefficient ring $R$, return the matrix with $(r+s)$ rows and $c$ columns over $R$ obtained by joining $X$ and $Y$ vertically (placing $Y$ underneath $X$).

## `VerticalJoin(Q): [ Mtrx ] -> Mtrx`

## `VerticalJoin(T): < Mtrx > -> Mtrx`

Given a sequence $Q$ or tuple $T$ of matrices, each having the same number of columns and being over the same coefficient ring $R$, return the matrix over $R$ obtained by joining the elements of $Q$ or $T$ vertically in order.

## `DiagonalJoin(X, Y): Mtrx, Mtrx -> Mtrx`

Given matrices $X$ with $a$ rows and $b$ columns and $Y$ with $c$ rows and $d$ columns, both over the same coefficient ring $R$, return the matrix with $(a+c)$ rows and $(b+d)$ columns over $R$ obtained by joining $X$ and $Y$ diagonally (placing $Y$ diagonally to the right of and underneath $X$, with zero blocks above and below the diagonal).

## `DiagonalJoin(Q): [ Mtrx ] -> Mtrx`

## `DiagonalJoin(T): < Mtrx > -> Mtrx`

Given a sequence $Q$ or tuple $T$ of matrices, each being over the same coefficient ring $R$, return the matrix over $R$ obtained by joining the elements of $Q$ or $T$ diagonally in order.

## `KroneckerProduct(A, B): Mtrx, Mtrx -> Mtrx`

Given an $m \times n$ matrix $A$ and a $p \times q$ matrix $B$, both over a ring $R$, return the Kronecker product of $A$ and $B$, which is the $mp\times nq$ matrix $C$ over $R$ such that the $((i-1)p+r, (j-1)q+s)$-th entry of $C$ is the $(i,j)$-th entry of $A$ times the $(r,s)$-th entry of $B$, for $1\leq i\le m$, $1\leq j\leq n$, $1\leq r\leq p$ and $1\leq s\leq q$.