The '.' operator performs noncommutative or dot product multiplication on its arguments.

If A and B are numbers (including complex and extended numerics such as infinity and undefined), then A . B = A*B.

If A and B are Vectors with the same orientation (i.e., both are row Vectors or both are column Vectors), then A . B is computed by using LinearAlgebra[DotProduct].

If one of A and B is a Matrix or a Vector, and the other is a Matrix, Vector or constant and the previous case does not apply, then their product is computed by using LinearAlgebra[Multiply].  See also simplify/rtable.  

If A and B are Arrays, their product is computed as component-wise multiplication using zip.  If A and B do not have the same dimensions, extra entries are ignored.

Arguments that are not of type Matrix, Vector, constant, or Array are ignored, and A . B remains unevaluated.

Calls to '.' such as A . B . C call `.`(A, B, C).

In this case, the process is repeated on the (transformed) arguments until no such argument pairs remain.

The dot operator is left-associative.

Note:  In Maple,  '.' can be interpreted as a decimal point (for example, $3.7$), as part of a range operator (for example, $x..y$), or as the (noncommutative) multiplication operator.  To distinguish between these three circumstances, Maple uses the following rule.

Any dot that is not part of a range operator (more than one '.' in a row) and not part of a number is interpreted as the noncommutative multiplication operator.

Note that the interpretation of the phrase "not part of a number" depends on whether you are using 1-D or 2-D input mode.  In 1-D input mode, interpretation proceeds from left to right, and a dot following a number will be interpreted as a decimal point unless that number already contains a decimal point.  In 2-D input mode, interpretation is carried out on the expression as a whole, and because spaces and juxtaposition can be interpreted as multiplication, a dot which is immediately preceded or followed by a number is always interpreted as a decimal point.

For example, in 1-D input mode, 3.4 is a number, 3. 4 is an error and 3 .4 and 3 . 4 return 12.  3. .4 is 12. and 3..4 is a range.

In 2-D input mode, 3.4 is a number, 3. 4 and 3 .4 are errors and 3 . 4 returns 12.  3. .4 is an error and 3..4 is again a range.  (All of the errors shown by these examples are due to the rule that a number cannot appear as the right-hand operand of an implicit multiplication operation. In such cases, use of explicit multiplication ( * ) can avoid the error.  See also 2-D Math Details for more information.)

The `.` operator is thread-safe as of Maple 15.

For more information on thread safety, see index/threadsafe.

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

$A≔\mathrm{ScalarMatrix}\left(3\,3\right)$

$A≔\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$

$B≔⟨⟨1\,0\,2⟩\|⟨0\,1\,2⟩\|⟨0\,0\,2⟩⟩$

$B≔\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 2& 2& 2\end{array}\right]$

$V≔⟨-1\,2\,-5⟩$

$V≔\left[\begin{array}{c}\mathrm{-1}\\ 2\\ \mathrm{-5}\end{array}\right]$

$\mathrm{`.`}\left(A\,B\,V\right)$

$\left[\begin{array}{c}\mathrm{-3}\\ 6\\ \mathrm{-24}\end{array}\right]$

$\mathrm{`.`}\left(V\,V\right)$

$30$

$\mathrm{`.`}\left(B\,\mathrm{`.`}\left(V\,V\right)\right)$

$\left[\begin{array}{ccc}30& 0& 0\\ 0& 30& 0\\ 60& 60& 60\end{array}\right]$

$\mathrm{`.`}\left(B\,V\,V\right)$

$45$

$\mathrm{`.`}\left(4\,V\right)$

$\left[\begin{array}{c}\mathrm{-4}\\ 8\\ \mathrm{-20}\end{array}\right]$

$\mathrm{`.`}\left(\mathrm{\λ}\,A\,B\right)$

$\mathrm{\lambda }·\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 6& 6& 6\end{array}\right]$

$\mathrm{`.`}\left(a\,b\right)$

$a·b$

$\mathrm{`.`}\left(7\,6\right)$

$42$

$\mathrm{Array1}≔\mathrm{Array}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\,6\right]\right]\right)$

$\mathrm{Array1}≔\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$

$\mathrm{Array2}≔\mathrm{Array}\left(\left[\left[1\,1\,1\right]\,\left[2\,2\,2\right]\,\left[3\,3\,3\right]\right]\right)$

$\mathrm{Array2}≔\left[\begin{array}{ccc}1& 1& 1\\ 2& 2& 2\\ 3& 3& 3\end{array}\right]$

$\mathrm{Array3}≔\mathrm{Array}\left(\left[\left[1\,2\,3\right]\,\left[1\,2\,3\right]\,\left[1\,2\,3\right]\right]\right)$

$\mathrm{Array3}≔\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\end{array}\right]$

$\mathrm{`.`}\left(\mathrm{Array1}\,\mathrm{Array2}\right)$

$\left[\begin{array}{ccc}1& 2& 3\\ 8& 10& 12\end{array}\right]$

$\mathrm{`.`}\left(\mathrm{Array2}\,\mathrm{Array3}\right)$

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$