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( T o p )
1
A b s t r a c t d e f i n i t i o n
2
D e f i n i t i o n i n c o o r d i n a t e s
T o g g l e D e f i n i t i o n i n c o o r d i n a t e s s u b s e c t i o n
2 . 1
E l e m e n t - w i s e d e f i n i t i o n
3
P r o p e r t i e s
4
C o m p u t a t i o n
5
A p p l i c a t i o n s
6
F u r t h e r r e a d i n g
T o g g l e t h e t a b l e o f c o n t e n t s
M u l t i l i n e a r m u l t i p l i c a t i o n
A d d l a n g u a g e s
A d d l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Let
F
{\displaystyle F}
be a field of characteristic zero, such as
R
{\displaystyle \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
.
Let
V
k
{\displaystyle V_{k}}
be a finite-dimensional vector space over
F
{\displaystyle F}
, and let
A
∈
V
1
⊗
V
2
⊗
⋯
⊗
V
d
{\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}}
be an order-d simple tensor , i.e., there exist some vectors
v
k
∈
V
k
{\displaystyle \mathbf {v} _{k}\in V_{k}}
such that
A
=
v
1
⊗
v
2
⊗
⋯
⊗
v
d
{\displaystyle {\mathcal {A}}=\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}}
. If we are given a collection of linear maps
A
k
:
V
k
→
W
k
{\displaystyle A_{k}:V_{k}\to W_{k}}
, then the multilinear multiplication of
A
{\displaystyle {\mathcal {A}}}
with
(
A
1
,
A
2
,
…
,
A
d
)
{\displaystyle (A_{1},A_{2},\ldots ,A_{d})}
is defined[1] as the action on
A
{\displaystyle {\mathcal {A}}}
of the tensor product of these linear maps,[2] namely
A
1
⊗
A
2
⊗
⋯
⊗
A
d
:
V
1
⊗
V
2
⊗
⋯
⊗
V
d
→
W
1
⊗
W
2
⊗
⋯
⊗
W
d
,
v
1
⊗
v
2
⊗
⋯
⊗
v
d
↦
A
1
(
v
1
)
⊗
A
2
(
v
2
)
⊗
⋯
⊗
A
d
(
v
d
)
{\displaystyle {\begin{aligned}A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d}:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}&\to W_{1}\otimes W_{2}\otimes \cdots \otimes W_{d},\\\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}&\mapsto A_{1}(\mathbf {v} _{1})\otimes A_{2}(\mathbf {v} _{2})\otimes \cdots \otimes A_{d}(\mathbf {v} _{d})\end{aligned}}}
Since the tensor product of linear maps is itself a linear map,[2] and because every tensor admits a tensor rank decomposition ,[1] the above expression extends linearly to all tensors. That is, for a general tensor
A
∈
V
1
⊗
V
2
⊗
⋯
⊗
V
d
{\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}}
, the multilinear multiplication is
B
:=
(
A
1
⊗
A
2
⊗
⋯
⊗
A
d
)
(
A
)
=
(
A
1
⊗
A
2
⊗
⋯
⊗
A
d
)
(
∑
i
=
1
r
a
i
1
⊗
a
i
2
⊗
⋯
⊗
a
i
d
)
=
∑
i
=
1
r
A
1
(
a
i
1
)
⊗
A
2
(
a
i
2
)
⊗
⋯
⊗
A
d
(
a
i
d
)
{\displaystyle {\begin{aligned}&{\mathcal {B}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})\\[4pt]={}&(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})\left(\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)\\[5pt]={}&\sum _{i=1}^{r}A_{1}(\mathbf {a} _{i}^{1})\otimes A_{2}(\mathbf {a} _{i}^{2})\otimes \cdots \otimes A_{d}(\mathbf {a} _{i}^{d})\end{aligned}}}
where
A
=
∑
i
=
1
r
a
i
1
⊗
a
i
2
⊗
⋯
⊗
a
i
d
{\textstyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}}
with
a
i
k
∈
V
k
{\displaystyle \mathbf {a} _{i}^{k}\in V_{k}}
is one of
A
{\displaystyle {\mathcal {A}}}
's tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of
A
{\displaystyle {\mathcal {A}}}
as a linear combination of pure tensors, which follows from the universal property of the tensor product .
It is standard to use the following shorthand notations in the literature for multilinear multiplications:
(
A
1
,
A
2
,
…
,
A
d
)
⋅
A
:=
(
A
1
⊗
A
2
⊗
⋯
⊗
A
d
)
(
A
)
{\displaystyle (A_{1},A_{2},\ldots ,A_{d})\cdot {\mathcal {A}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})}
and
A
k
⋅
k
A
:=
(
Id
V
1
,
…
,
Id
V
k
−
1
,
A
k
,
Id
V
k
+
1
,
…
,
Id
V
d
)
⋅
A
,
{\displaystyle A_{k}\cdot _{k}{\mathcal {A}}:=(\operatorname {Id} _{V_{1}},\ldots ,\operatorname {Id} _{V_{k-1}},A_{k},\operatorname {Id} _{V_{k+1}},\ldots ,\operatorname {Id} _{V_{d}})\cdot {\mathcal {A}},}
where
Id
V
k
:
V
k
→
V
k
{\displaystyle \operatorname {Id} _{V_{k}}:V_{k}\to V_{k}}
is the identity operator .
Definition in coordinates [ edit ]
In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on
V
k
{\displaystyle V_{k}}
and let
V
k
∗
{\displaystyle V_{k}^{*}}
denote the dual vector space of
V
k
{\displaystyle V_{k}}
. Let
{
e
1
k
,
…
,
e
n
k
k
}
{\displaystyle \{e_{1}^{k},\ldots ,e_{n_{k}}^{k}\}}
be a basis for
V
k
{\displaystyle V_{k}}
, let
{
(
e
1
k
)
∗
,
…
,
(
e
n
k
k
)
∗
}
{\displaystyle \{(e_{1}^{k})^{*},\ldots ,(e_{n_{k}}^{k})^{*}\}}
be the dual basis, and let
{
f
1
k
,
…
,
f
m
k
k
}
{\displaystyle \{f_{1}^{k},\ldots ,f_{m_{k}}^{k}\}}
be a basis for
W
k
{\displaystyle W_{k}}
. The linear map
M
k
=
∑
i
=
1
m
k
∑
j
=
1
n
k
m
i
,
j
(
k
)
f
i
k
⊗
(
e
j
k
)
∗
{\textstyle M_{k}=\sum _{i=1}^{m_{k}}\sum _{j=1}^{n_{k}}m_{i,j}^{(k )}f_{i}^{k}\otimes (e_{j}^{k})^{*}}
is then represented by the matrix
M
^
k
=
[
m
i
,
j
(
k
)
]
∈
F
m
k
×
n
k
{\displaystyle {\widehat {M}}_{k}=[m_{i,j}^{(k )}]\in F^{m_{k}\times n_{k}}}
. Likewise, with respect to the standard tensor product basis
{
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
}
j
1
,
j
2
,
…
,
j
d
{\displaystyle \{e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}\}_{j_{1},j_{2},\ldots ,j_{d}}}
, the abstract tensor
A
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
{\displaystyle {\mathcal {A}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}}
is represented by the multidimensional array
A
^
=
[
a
j
1
,
j
2
,
…
,
j
d
]
∈
F
n
1
×
n
2
×
⋯
×
n
d
{\displaystyle {\widehat {\mathcal {A}}}=[a_{j_{1},j_{2},\ldots ,j_{d}}]\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}}
. Observe that
A
^
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
,
{\displaystyle {\widehat {\mathcal {A}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},}
where
e
j
k
∈
F
n
k
{\displaystyle \mathbf {e} _{j}^{k}\in F^{n_{k}}}
is the j th standard basis vector of
F
n
k
{\displaystyle F^{n_{k}}}
and the tensor product of vectors is the affine Segre map
⊗
:
(
v
(
1
)
,
v
(
2
)
,
…
,
v
(
d
)
)
↦
[
v
i
1
(
1
)
v
i
2
(
2
)
⋯
v
i
d
(
d
)
]
i
1
,
i
2
,
…
,
i
d
{\displaystyle \otimes :(\mathbf {v} ^{(1 )},\mathbf {v} ^{(2 )},\ldots ,\mathbf {v} ^{(d )})\mapsto [v_{i_{1}}^{(1 )}v_{i_{2}}^{(2 )}\cdots v_{i_{d}}^{(d )}]_{i_{1},i_{2},\ldots ,i_{d}}}
. It follows from the above choices of bases that the multilinear multiplication
B
=
(
M
1
,
M
2
,
…
,
M
d
)
⋅
A
{\displaystyle {\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}}
becomes
B
^
=
(
M
^
1
,
M
^
2
,
…
,
M
^
d
)
⋅
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
(
M
^
1
,
M
^
2
,
…
,
M
^
d
)
⋅
(
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
)
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
(
M
^
1
e
j
1
1
)
⊗
(
M
^
2
e
j
2
2
)
⊗
⋯
⊗
(
M
^
d
e
j
d
d
)
.
{\displaystyle {\begin{aligned}{\widehat {\mathcal {B}}}&=({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot (\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d})\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ({\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ({\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d}).\end{aligned}}}
The resulting tensor
B
^
{\displaystyle {\widehat {\mathcal {B}}}}
lives in
F
m
1
×
m
2
×
⋯
×
m
d
{\displaystyle F^{m_{1}\times m_{2}\times \cdots \times m_{d}}}
.
Element-wise definition [ edit ]
From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since
B
^
{\displaystyle {\widehat {\mathcal {B}}}}
is a multidimensional array, it may be expressed as
B
^
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
b
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
,
{\displaystyle {\widehat {\mathcal {B}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},}
where
b
j
1
,
j
2
,
…
,
j
d
∈
F
{\displaystyle b_{j_{1},j_{2},\ldots ,j_{d}}\in F}
are the coefficients. Then it follows from the above formulae that
(
(
e
i
1
1
)
T
,
(
e
i
2
2
)
T
,
…
,
(
e
i
d
d
)
T
)
⋅
B
^
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
b
j
1
,
j
2
,
…
,
j
d
(
(
e
i
1
1
)
T
e
j
1
1
)
⊗
(
(
e
i
2
2
)
T
e
j
2
2
)
⊗
⋯
⊗
(
(
e
i
d
d
)
T
e
j
d
d
)
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
b
j
1
,
j
2
,
…
,
j
d
δ
i
1
,
j
1
⋅
δ
i
2
,
j
2
⋯
δ
i
d
,
j
d
=
b
i
1
,
i
2
,
…
,
i
d
,
{\displaystyle {\begin{aligned}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\left((\mathbf {e} _{i_{1}}^{1})^{T}\mathbf {e} _{j_{1}}^{1}\right)\otimes \left((\mathbf {e} _{i_{2}}^{2})^{T}\mathbf {e} _{j_{2}}^{2}\right)\otimes \cdots \otimes \left((\mathbf {e} _{i_{d}}^{d})^{T}\mathbf {e} _{j_{d}}^{d}\right)\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\delta _{i_{1},j_{1}}\cdot \delta _{i_{2},j_{2}}\cdots \delta _{i_{d},j_{d}}\\={}&b_{i_{1},i_{2},\ldots ,i_{d}},\end{aligned}}}
where
δ
i
,
j
{\displaystyle \delta _{i,j}}
is the Kronecker delta . Hence, if
B
=
(
M
1
,
M
2
,
…
,
M
d
)
⋅
A
{\displaystyle {\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}}
, then
b
i
1
,
i
2
,
…
,
i
d
=
(
(
e
i
1
1
)
T
,
(
e
i
2
2
)
T
,
…
,
(
e
i
d
d
)
T
)
⋅
B
^
=
(
(
e
i
1
1
)
T
,
(
e
i
2
2
)
T
,
…
,
(
e
i
d
d
)
T
)
⋅
(
M
^
1
,
M
^
2
,
…
,
M
^
d
)
⋅
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
(
(
e
i
1
1
)
T
M
^
1
e
j
1
1
)
⊗
(
(
e
i
2
2
)
T
M
^
2
e
j
2
2
)
⊗
⋯
⊗
(
(
e
i
d
d
)
T
M
^
d
e
j
d
d
)
=
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
m
i
1
,
j
1
(
1
)
⋅
m
i
2
,
j
2
(
2
)
⋯
m
i
d
,
j
d
(
d
)
,
{\displaystyle {\begin{aligned}&b_{i_{1},i_{2},\ldots ,i_{d}}=\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot ({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}((\mathbf {e} _{i_{1}}^{1})^{T}{\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ((\mathbf {e} _{i_{2}}^{2})^{T}{\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ((\mathbf {e} _{i_{d}}^{d})^{T}{\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d})\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}m_{i_{1},j_{1}}^{(1 )}\cdot m_{i_{2},j_{2}}^{(2 )}\cdots m_{i_{d},j_{d}}^{(d )},\end{aligned}}}
where the
m
i
,
j
(
k
)
{\displaystyle m_{i,j}^{(k )}}
are the elements of
M
^
k
{\displaystyle {\widehat {M}}_{k}}
as defined above.
Properties [ edit ]
Let
A
∈
V
1
⊗
V
2
⊗
⋯
⊗
V
d
{\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}}
be an order-d tensor over the tensor product of
F
{\displaystyle F}
-vector spaces.
Since a multilinear multiplication is the tensor product of linear maps, we have the following multilinearity property (in the construction of the map):[1] [2]
A
1
⊗
⋯
⊗
A
k
−
1
⊗
(
α
A
k
+
β
B
)
⊗
A
k
+
1
⊗
⋯
⊗
A
d
=
α
A
1
⊗
⋯
⊗
A
d
+
β
A
1
⊗
⋯
⊗
A
k
−
1
⊗
B
⊗
A
k
+
1
⊗
⋯
⊗
A
d
{\displaystyle A_{1}\otimes \cdots \otimes A_{k-1}\otimes (\alpha A_{k}+\beta B)\otimes A_{k+1}\otimes \cdots \otimes A_{d}=\alpha A_{1}\otimes \cdots \otimes A_{d}+\beta A_{1}\otimes \cdots \otimes A_{k-1}\otimes B\otimes A_{k+1}\otimes \cdots \otimes A_{d}}
Multilinear multiplication is a linear map :[1] [2]
(
M
1
,
M
2
,
…
,
M
d
)
⋅
(
α
A
+
β
B
)
=
α
(
M
1
,
M
2
,
…
,
M
d
)
⋅
A
+
β
(
M
1
,
M
2
,
…
,
M
d
)
⋅
B
{\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot (\alpha {\mathcal {A}}+\beta {\mathcal {B}})=\alpha \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}+\beta \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {B}}}
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:[1] [2]
(
M
1
,
M
2
,
…
,
M
d
)
⋅
(
(
K
1
,
K
2
,
…
,
K
d
)
⋅
A
)
=
(
M
1
∘
K
1
,
M
2
∘
K
2
,
…
,
M
d
∘
K
d
)
⋅
A
,
{\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot \left((K_{1},K_{2},\ldots ,K_{d})\cdot {\mathcal {A}}\right)=(M_{1}\circ K_{1},M_{2}\circ K_{2},\ldots ,M_{d}\circ K_{d})\cdot {\mathcal {A}},}
where
M
k
:
U
k
→
W
k
{\displaystyle M_{k}:U_{k}\to W_{k}}
and
K
k
:
V
k
→
U
k
{\displaystyle K_{k}:V_{k}\to U_{k}}
are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
M
k
⋅
k
(
M
ℓ
⋅
ℓ
A
)
=
M
ℓ
⋅
ℓ
(
M
k
⋅
k
A
)
=
M
k
⋅
k
M
ℓ
⋅
ℓ
A
,
{\displaystyle M_{k}\cdot _{k}\left(M_{\ell }\cdot _{\ell }{\mathcal {A}}\right)=M_{\ell }\cdot _{\ell }\left(M_{k}\cdot _{k}{\mathcal {A}}\right)=M_{k}\cdot _{k}M_{\ell }\cdot _{\ell }{\mathcal {A}},}
if
k
≠
ℓ
.
{\displaystyle k\neq \ell .}
Computation [ edit ]
The factor-k multilinear multiplication
M
k
⋅
k
A
{\displaystyle M_{k}\cdot _{k}{\mathcal {A}}}
can be computed in coordinates as follows. Observe first that
M
k
⋅
k
A
=
M
k
⋅
k
∑
j
1
=
1
n
1
∑
j
2
=
1
n
2
⋯
∑
j
d
=
1
n
d
a
j
1
,
j
2
,
…
,
j
d
e
j
1
1
⊗
e
j
2
2
⊗
⋯
⊗
e
j
d
d
=
∑
j
1
=
1
n
1
⋯
∑
j
k
−
1
=
1
n
k
−
1
∑
j
k
+
1
=
1
n
k
+
1
⋯
∑
j
d
=
1
n
d
e
j
1
1
⊗
⋯
⊗
e
j
k
−
1
k
−
1
⊗
M
k
(
∑
j
k
=
1
n
k
a
j
1
,
j
2
,
…
,
j
d
e
j
k
k
)
⊗
e
j
k
+
1
k
+
1
⊗
⋯
⊗
e
j
d
d
.
{\displaystyle {\begin{aligned}M_{k}\cdot _{k}{\mathcal {A}}&=M_{k}\cdot _{k}\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \cdots \otimes \mathbf {e} _{j_{k-1}}^{k-1}\otimes M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{j_{k+1}}^{k+1}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}.\end{aligned}}}
Next, since
F
n
1
⊗
F
n
2
⊗
⋯
⊗
F
n
d
≃
F
n
k
⊗
(
F
n
1
⊗
⋯
⊗
F
n
k
−
1
⊗
F
n
k
+
1
⊗
⋯
⊗
F
n
d
)
≃
F
n
k
⊗
F
n
1
⋯
n
k
−
1
n
k
+
1
⋯
n
d
,
{\displaystyle F^{n_{1}}\otimes F^{n_{2}}\otimes \cdots \otimes F^{n_{d}}\simeq F^{n_{k}}\otimes (F^{n_{1}}\otimes \cdots \otimes F^{n_{k-1}}\otimes F^{n_{k+1}}\otimes \cdots \otimes F^{n_{d}})\simeq F^{n_{k}}\otimes F^{n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}},}
there is a bijective map, called the factor-k standard flattening ,[1] denoted by
(
⋅
)
(
k
)
{\displaystyle (\cdot )_{(k )}}
, that identifies
M
k
⋅
k
A
{\displaystyle M_{k}\cdot _{k}{\mathcal {A}}}
with an element from the latter space, namely
(
M
k
⋅
k
A
)
(
k
)
:=
∑
j
1
=
1
n
1
⋯
∑
j
k
−
1
=
1
n
k
−
1
∑
j
k
+
1
=
1
n
k
+
1
⋯
∑
j
d
=
1
n
d
M
k
(
∑
j
k
=
1
n
k
a
j
1
,
j
2
,
…
,
j
d
e
j
k
k
)
⊗
e
μ
k
(
j
1
,
…
,
j
k
−
1
,
j
k
+
1
,
…
,
j
d
)
:=
M
k
A
(
k
)
,
{\displaystyle \left(M_{k}\cdot _{k}{\mathcal {A}}\right)_{(k )}:=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{\mu _{k}(j_{1},\ldots ,j_{k-1},j_{k+1},\ldots ,j_{d})}:=M_{k}{\mathcal {A}}_{(k )},}
where
e
j
{\displaystyle \mathbf {e} _{j}}
is the j th standard basis vector of
F
N
k
{\displaystyle F^{N_{k}}}
,
N
k
=
n
1
⋯
n
k
−
1
n
k
+
1
⋯
n
d
{\displaystyle N_{k}=n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}}
, and
A
(
k
)
∈
F
n
k
⊗
F
N
k
≃
F
n
k
×
N
k
{\displaystyle {\mathcal {A}}_{(k )}\in F^{n_{k}}\otimes F^{N_{k}}\simeq F^{n_{k}\times N_{k}}}
is the factor-k flattening matrix of
A
{\displaystyle {\mathcal {A}}}
whose columns are the factor-k vectors
[
a
j
1
,
…
,
j
k
−
1
,
i
,
j
k
+
1
,
…
,
j
d
]
i
=
1
n
k
{\displaystyle [a_{j_{1},\ldots ,j_{k-1},i,j_{k+1},\ldots ,j_{d}}]_{i=1}^{n_{k}}}
in some order, determined by the particular choice of the bijective map
μ
k
:
[
1
,
n
1
]
×
⋯
×
[
1
,
n
k
−
1
]
×
[
1
,
n
k
+
1
]
×
⋯
×
[
1
,
n
d
]
→
[
1
,
N
k
]
.
{\displaystyle \mu _{k}:[1,n_{1}]\times \cdots \times [1,n_{k-1}]\times [1,n_{k+1}]\times \cdots \times [1,n_{d}]\to [1,N_{k}].}
In other words, the multilinear multiplication
(
M
1
,
M
2
,
…
,
M
d
)
⋅
A
{\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}}
can be computed as a sequence of d factor-k multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.
Applications [ edit ]
The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates
A
∈
F
n
1
×
n
2
×
⋯
×
n
d
{\displaystyle {\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}}
as the multilinear multiplication
A
=
(
U
1
,
U
2
,
…
,
U
d
)
⋅
S
{\displaystyle {\mathcal {A}}=(U_{1},U_{2},\ldots ,U_{d})\cdot {\mathcal {S}}}
, where
U
k
∈
F
n
k
×
n
k
{\displaystyle U_{k}\in F^{n_{k}\times n_{k}}}
are orthogonal matrices and
S
∈
F
n
1
×
n
2
×
⋯
×
n
d
{\displaystyle {\mathcal {S}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}}
.
Further reading [ edit ]
^ a b c d e Multilinear Algebra | Werner Greub | Springer . Universitext. Springer. 1978. ISBN 9780387902845 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Multilinear_multiplication&oldid=955077490 "
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● T e n s o r s
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● T h i s p a g e w a s l a s t e d i t e d o n 5 M a y 2 0 2 0 , a t 2 1 : 0 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
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