In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
f
:
V
1
×
⋯
×
V
n
→
W
,
where
V
1
,
…
,
V
n
and
W
are vector spaces (or modules over a commutative ring), with the following property: for each
i
, if all of the variables but
v
i
are held constant, then
f
(
v
1
,
…
,
v
n
)
is a linear function of
v
i
.
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in
R
3
.
The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix.
If
F
:
R
m
→
R
n
is a Ck function, then the
k
th derivative of
F
at each point
p
in its domain can be viewed as a symmetric
k
-linear function
D
k
f
(
p
)
:
R
m
×
⋯
×
R
m
→
R
n
.
The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.
Let
f
:
V
1
×
⋯
×
V
n
→
W
,
be a multilinear map between finite-dimensional vector spaces, where
V
i
has dimension
d
i
, and
W
has dimension
d
. If we choose a basis
{
e
i
1
,
…
,
e
i
d
i
}
for each
V
i
and a basis
{
b
1
,
…
,
b
d
}
for
W
(using bold for vectors), then we can define a collection of scalars
A
j
1
⋯
j
n
k
by
f
(
e
1
j
1
,
…
,
e
n
j
n
)
=
A
j
1
⋯
j
n
1
b
1
+
⋯
+
A
j
1
⋯
j
n
d
b
d
.
Then the scalars
{
A
j
1
⋯
j
n
k
∣
1
≤
j
i
≤
d
i
,
1
≤
k
≤
d
}
completely determine the multilinear function
f
. In particular, if
v
i
=
∑
j
=
1
d
i
v
i
j
e
i
j
for
1
≤
i
≤
n
, then
f
(
v
1
,
…
,
v
n
)
=
∑
j
1
=
1
d
1
⋯
∑
j
n
=
1
d
n
∑
k
=
1
d
A
j
1
⋯
j
n
k
v
1
j
1
⋯
v
n
j
n
b
k
.
Let's take a trilinear function
f
:
R
2
×
R
2
×
R
2
→
R
,
where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
A basis for each Vi is
{
e
i
1
,
…
,
e
i
d
i
}
=
{
e
1
,
e
2
}
=
{
(
1
,
0
)
,
(
0
,
1
)
}
.
Let
f
(
e
1
i
,
e
2
j
,
e
3
k
)
=
f
(
e
i
,
e
j
,
e
k
)
=
A
i
j
k
,
where
i
,
j
,
k
∈
{
1
,
2
}
. In other words, the constant
A
i
j
k
is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three
V
i
), namely:
{
e
1
,
e
1
,
e
1
}
,
{
e
1
,
e
1
,
e
2
}
,
{
e
1
,
e
2
,
e
1
}
,
{
e
1
,
e
2
,
e
2
}
,
{
e
2
,
e
1
,
e
1
}
,
{
e
2
,
e
1
,
e
2
}
,
{
e
2
,
e
2
,
e
1
}
,
{
e
2
,
e
2
,
e
2
}
.
Each vector
v
i
∈
V
i
=
R
2
can be expressed as a linear combination of the basis vectors
v
i
=
∑
j
=
1
2
v
i
j
e
i
j
=
v
i
1
×
e
1
+
v
i
2
×
e
2
=
v
i
1
×
(
1
,
0
)
+
v
i
2
×
(
0
,
1
)
.
The function value at an arbitrary collection of three vectors
v
i
∈
R
2
can be expressed as
f
(
v
1
,
v
2
,
v
3
)
=
∑
i
=
1
2
∑
j
=
1
2
∑
k
=
1
2
A
i
j
k
v
1
i
v
2
j
v
3
k
.
Or, in expanded form as
f
(
(
a
,
b
)
,
(
c
,
d
)
,
(
e
,
f
)
)
=
a
c
e
×
f
(
e
1
,
e
1
,
e
1
)
+
a
c
f
×
f
(
e
1
,
e
1
,
e
2
)
+
a
d
e
×
f
(
e
1
,
e
2
,
e
1
)
+
a
d
f
×
f
(
e
1
,
e
2
,
e
2
)
+
b
c
e
×
f
(
e
2
,
e
1
,
e
1
)
+
b
c
f
×
f
(
e
2
,
e
1
,
e
2
)
+
b
d
e
×
f
(
e
2
,
e
2
,
e
1
)
+
b
d
f
×
f
(
e
2
,
e
2
,
e
2
)
.
There is a natural one-to-one correspondence between multilinear maps
f
:
V
1
×
⋯
×
V
n
→
W
,
and linear maps
F
:
V
1
⊗
⋯
⊗
V
n
→
W
,
where
V
1
⊗
⋯
⊗
V
n
denotes the tensor product of
V
1
,
…
,
V
n
. The relation between the functions
f
and
F
is given by the formula
F
(
v
1
⊗
⋯
⊗
v
n
)
=
f
(
v
1
,
…
,
v
n
)
.
One can consider multilinear functions, on an n×n matrix over a commutative ring K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
D
(
A
)
=
D
(
a
1
,
…
,
a
n
)
,
satisfying
D
(
a
1
,
…
,
c
a
i
+
a
i
′
,
…
,
a
n
)
=
c
D
(
a
1
,
…
,
a
i
,
…
,
a
n
)
+
D
(
a
1
,
…
,
a
i
′
,
…
,
a
n
)
.
If we let
e
^
j
represent the jth row of the identity matrix, we can express each row ai as the sum
a
i
=
∑
j
=
1
n
A
(
i
,
j
)
e
^
j
.
Using the multilinearity of D we rewrite D(A) as
D
(
A
)
=
D
(
∑
j
=
1
n
A
(
1
,
j
)
e
^
j
,
a
2
,
…
,
a
n
)
=
∑
j
=
1
n
A
(
1
,
j
)
D
(
e
^
j
,
a
2
,
…
,
a
n
)
.
Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,
D
(
A
)
=
∑
1
≤
k
i
≤
n
A
(
1
,
k
1
)
A
(
2
,
k
2
)
…
A
(
n
,
k
n
)
D
(
e
^
k
1
,
…
,
e
^
k
n
)
,
where, since in our case 1 ≤ i ≤ n,
is a series of nested summations.
Therefore, D(A) is uniquely determined by how D operates on
e
^
k
1
,
…
,
e
^
k
n
.
In the case of 2×2 matrices we get
D
(
A
)
=
A
1
,
1
A
2
,
1
D
(
e
^
1
,
e
^
1
)
+
A
1
,
1
A
2
,
2
D
(
e
^
1
,
e
^
2
)
+
A
1
,
2
A
2
,
1
D
(
e
^
2
,
e
^
1
)
+
A
1
,
2
A
2
,
2
D
(
e
^
2
,
e
^
2
)
Where
e
^
1
=
[
1
,
0
]
and
e
^
2
=
[
0
,
1
]
. If we restrict
D
to be an alternating function then
D
(
e
^
1
,
e
^
1
)
=
D
(
e
^
2
,
e
^
2
)
=
0
and
D
(
e
^
2
,
e
^
1
)
=
−
D
(
e
^
1
,
e
^
2
)
=
−
D
(
I
)
. Letting
D
(
I
)
=
1
we get the determinant function on 2×2 matrices:
D
(
A
)
=
A
1
,
1
A
2
,
2
−
A
1
,
2
A
2
,
1
.
A multilinear map has a value of zero whenever one of its arguments is zero.
