The Multiply function performs the operation A*B placing the output in a new object, or in C, depending on calling sequence. All Matrices or Vectors must have the same datatype. Multiply can perform Matrix-Matrix, Matrix-Vector, Vector-Matrix, scalar-Matrix, Vector-Vector, or scalar-Vector multiplication with output as a scalar,Vector or Matrix. The output depends on the input specification.

The first calling sequence returns a scalar or new mod m Matrix or Vector for the result with the specified ordering order, or the ordering of the input objects A and B.

Note: If A and B are both Matrices or Vectors and have a different ordering, order must be specified.

The remaining calling sequence places the output of the computation into C, returning NULL.

Scalar multiplication can be performed by setting A to a scalar value in the range $1..m-1$, and specifying B as a mod m Matrix or Vector. In this case, C can be specified as the same Matrix or Vector as B as long as the operation is performed with direct overlap. For example, if B and C are the same Matrix, the operation involving the entry $B_{i,j}$ must have the result going to $C_{i,j}$.

If the optional keyword, 'sparse', is specified, it only applies to scalar multiplication if the multiplier is not $1$ or $m-1$ and the datatype is a hardware datatype. It indicates that the algorithm checks for zero entries before performing multiplications. This typically provides an efficiency gain when matrices have 10% or more entries in B equal to zero. This is not implemented for the integer datatype, as it typically provides very little gain for that case.

Note: If sparse does not apply but it has been specified, the option is ignored.

For multiplication of two mod m Matrices or Vectors, the number of columns in A must be equal to the number of rows in B, and the result has dimensions equal to $\mathrm{rows}\left(A\right)x\mathrm{cols}\left(B\right)$.

For example, if A is a row Vector, and B is a column Vector, the result is a zero dimensional value representing the dot product of A and B. In this case, the output of the first calling sequence is a scalar value. For the second calling sequence, the entry in C into which this value is placed is determined by the sub-Matrix or sub-Vector specification for C.

Alternatively, if A is a column Vector, and B is a row Vector, the result is a Matrix which represents the outer product of A and B.

The Multiply function allows the use of sub-Matrix and sub-Vector specifications for both input and output Vectors/Matrices. For example, the function can be used to multiply a row of a Matrix to the transpose of a row in another Matrix, placing the result in a specified entry of a third Matrix.

Note: For scalar multiplication, B and C are allowed direct overlap. If both A and B are mod m Matrices or Vectors and if either overlap with C, the behavior of Multiply is undefined.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form Multiply(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][Multiply](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Modular}\right]\right)\:$

$A≔\mathrm{Mod}\left(13\,\mathrm{Vector}\left[\mathrm{row}\right]\left(4\,i\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$A≔\left[\begin{array}{cccc}10& 0& 8& 12\end{array}\right]$

$B≔\mathrm{Mod}\left(13\,\mathrm{Vector}\left[\mathrm{column}\right]\left(4\,i\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$B≔\left[\begin{array}{c}2\\ 6\\ 0\\ 11\end{array}\right]$

$\mathrm{Multiply}\left(13\,A\,B\right)$

$9$

$\mathrm{Multiply}\left(13\,B\,A\right)$

$\left[\begin{array}{cccc}7& 0& 3& 11\\ 8& 0& 9& 7\\ 0& 0& 0& 0\\ 6& 0& 10& 2\end{array}\right]$

$C≔\mathrm{Create}\left(13\,2\,2\,\mathrm{integer}\left[\right]\right)$

$C≔\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\mathrm{Multiply}\left(13\,A\,B\,C\,1\,2\right)\:$

$C$

$\left[\begin{array}{cc}0& 9\\ 0& 0\end{array}\right]$

$A≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(3\,4\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$A≔\left[\begin{array}{cccc}9& 2& 11& 7\\ 4& 11& 12& 11\\ 8& 9& 4& 11\end{array}\right]$

$\mathrm{Multiply}\left(13\,A\,B\right)$

$\left[\begin{array}{c}3\\ 0\\ 9\end{array}\right]$

$B≔\mathrm{Mod}\left(13\,\mathrm{Vector}\left[\mathrm{row}\right]\left(3\,i\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$B≔\left[\begin{array}{ccc}10& 3& 5\end{array}\right]$

$\mathrm{Multiply}\left(13\,B\,A\right)$

$\left[\begin{array}{cccc}12& 7& 10& 2\end{array}\right]$

$\mathrm{Multiply}\left(13\,2\,A\,A\right)\:$

$A$

$\left[\begin{array}{cccc}5& 4& 9& 1\\ 8& 9& 11& 9\\ 3& 5& 8& 9\end{array}\right]$

$C≔\mathrm{Create}\left(13\,2\,6\,\mathrm{integer}\left[\right]\right)$

$C≔\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$

$\mathrm{Multiply}\left(13\,A\,3\,A\,'\mathrm{transpose}'\,C\,2\,2..4\right)\:$

$C$

$\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 12& 4& 10& 0& 0\end{array}\right]$