⋅ (dot operator)
The multiplication operator.
Syntax

a ⋅ b

applicable to several different kinds of operands
a
andb

Description and examples
Multiplying two numbers
If a
and b
are numbers (integers, rational numbers, real number, or complex numbers), a ⋅ b
is the mathematical product of a
and b
. The type of the product is the most specific type possible.
Some examples:

If
a
andb
are both integers, so isa ⋅ b
if no integer overflow occurs; in that case, the result is a real number. 
If
a
is an integer andb
a rational number,a ⋅ b
is a rational number. 
If
a
is an integer andb
a real number,a ⋅ b
is a real number. 
If
a
is an integer andb
a complex number,a ⋅ b
is a complex number. 
If
a
is a real number andb
a complex number,a ⋅ b
is a complex number. 
If
a
is a rational number andb
a complex number,a ⋅ b
is a complex number.
√2⋅π
4.44288293816
3/7 ⋅ 2/5
6/35 (=0.171428571429)
Vector dot product
If u
and v
are two vectors of the same dimension, then u⋅v
is their dot (inner, scalar) product. If both are real, then u⋅v
is real. If any of u
and v
is complex, the complex inner inner product will be used and the result will be complex.
❨5, 0, 1❩ ⋅ ❨−2, 1, 0❩
−10
❨1, i, 3❩ ⋅ ❨2, 1, i❩
2 − 2⋅i
The vector dot product of u
and v
can also be written (uv)
or InnerProduct(u, v)
.
Matrix multiplication
If A
and B
are two matrices of compatible sizes, then A⋅B
is their matrix product. The number of columns of A
must equal the number of rows of B
; the result will have the same number of rows as A
and the same number of columns as B
.
If any of the operands is complex, so is the product; otherwise, the product is a real matrix.
A ≔ ❨❨3, 0, 1❩, ❨5, 1, 0❩, ❨0, 2, 1❩❩
⎛3 0 1⎞ ⎜5 1 0⎟ ⎝0 2 1⎠
B ≔ ❨❨3, 1, i❩, ❨2, 0, 1❩, ❨4, 3, −1❩❩
⎛ 3 1 i⎞ ⎜ 2 0 1⎟ ⎝ 4 3 −1⎠
A⋅B
⎛ 13 6 −1 + 3⋅i⎞ ⎜ 17 5 1 + 5⋅i⎟ ⎝ 8 3 1⎠
Multiplication by scalar
If v
is a vector, A
a matrix, and x
a number, then x⋅v
, v⋅x
, x⋅A
, and A⋅x
are the results of the corresponding multiplication by scalar operations, that is, each vector component or matrix entry is multiplied by the scalar, thus forming a new vector or matrix.
The result is complex iff at least one operand is complex.
2⋅❨1, 0, 0❩ + 3⋅❨0, 1, 0❩ + 5⋅❨0, 0, 1❩
⎛2⎞ e⎜3⎟ ⎝5⎠
5⋅IdentityMatrix(3)
⎛5 0 0⎞ ⎜0 5 0⎟ ⎝0 0 5⎠
Multiplication between matrix and vector
If A
is a matrix and v
a vector, with A
having the same number of columns as the dimension of v
, then A⋅v
is the image of v
under the linear transformation given by A
. In other words, A⋅v = A⋅matrix(v)
is the matrix product between A
and v
considered as a column vector (matrix).
The result is complex iff at least one operand is complex.
A ≔ ❨❨1, 0, 0❩, ❨0, 0, 1❩, ❨0, −1, 0❩❩
⎛ 1 0 0⎞ ⎜ 0 0 1⎟ ⎝ 0 −1 0⎠
A⋅❨1, 0, 2❩
⎛1⎞ e⎜2⎟ ⎝0⎠
String repetition
If n
is a nonnegative integer and s
a string, then n⋅s
or s⋅n
is the string s
repeated n
times.
10⋅"CTG"
CTGCTGCTGCTGCTGCTGCTGCTGCTGCTG
"━"⋅80
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Scaling a sound
If s
is a sound and x ∈ [0, 1]
, then x⋅s
is the sound obtained from s
by scaling the samples by x
. Hence, 1⋅s = s
and 0⋅s
is silence.
0.1 ⋅ SineTone(100, 1, 1)
A 1second 32bit 48000 Hz 1channel sound.
Combining two sounds
If s
and t
are two sounds, then s ⋅ t
is the same as 0.5⋅s + 0.5⋅t
.
SineTone(100, 0.1, 1) ⋅ SineTone(400, 0.1, 1)
A 1second 32bit 48000 Hz 1channel sound.
Notes
The binary operator ⋅
is mapped to the multiply
function.