Database Reference
In-Depth Information
Fig. 9.1 Illustration
of a tensor
A
ð
2
;
2
;
2
Þ
∈
R
0
0
0
0
-1
1
1
1
6. Multilinear p-mode product with a matrix
Special case of 5 for
δ ¼
1 and matrix
b
ð
Þ
R
ðÞ
!
R
ð
Þ
,
m
1
;...;
m
p
1
;
m
p
;
m
pþ
1
;...;
m
d
n
;
m
p
m
1
;...;
m
p
1
;
m
p
;
m
p
1
;...;
m
d
p
:
R
!
!
m
p
X
b
jk
j;ð ∈
n
aðÞ
p
b
jk
:¼
aðÞ
i
∈
m
;
a
i
1
...i
p
...i
d
b
ji
p
:
;
m
p
i
p
i
;ð ∈
m
ð
;
n
Þ
n
(w.r.t. the previously defined sum
and scalar multiplication) with the
Euclidean inner product
Furthermore, we endow the vector space
R
i :¼
X
i
∈
n
h
A
;
B
a
i
b
i
and the thus induced (generalized)
Frobenius norm
2
F
kk
:¼ A
h
;
A
i:
n
d
. We define n
(
k
)
Definition 9.2 Let
A
∈
R
k
∈
:
¼
(
n
1
,
...
,
n
k
1
,
n
k
+1
,
...
,
n
d
).
,
be 1-1,
i.e.
, an enumeration. Then the matrix
A
(
k
)
n
ðÞ
!
n
ðÞ
Furthermore, let
υ :
with entries
a
ðÞ
i
k
υ
ðÞ
:¼ a
i
, i
∈
n
i
ðÞ
is referred to as k-mode matricization of A.
As regards the scope of this chapter, the choice of enumeration is inconsequen-
tial if deployed consistently. Hence, in what follows, we shall consider the
k
-mode
matricization of a given tensor as a uniquely defined object.
ð
2
;
2
;
2
Þ
with entries
Example 9.1
Let
A
∈
R
a
1
;
1
;
1
Þ
¼
1,
a
1
;
2
;
1
Þ
¼
1,
a
1
;
1
;
2
Þ
¼
0,
a
1
;
2
;
2
Þ
¼
0,
ð
ð
ð
ð
a
2
;
1
;
1
Þ
¼
1,
a
2
;
2
;
1
Þ
¼
1,
a
2
;
1
;
2
Þ
¼
0,
a
2
;
2
;
2
Þ
¼
0
:
ð
ð
ð
ð
A graphical rendition of this tensor is provided by Fig.
9.1
.
