Database Reference
In-Depth Information
T
,
20
02
A
ðÞ
A
ðÞ
¼
and
T
40
00
A
ðÞ
A
ðÞ
¼
:
Hence,
U
1
¼ U
2
¼ U
3
¼ I
. Since there are multiple eigenvalues in the first two
modes, there is no unique (1,1,2) truncated HOSVD. Instead, any of the subspaces
spanned by
e
1
or
e
2
are 1-dimensional principal subspaces of both of the matricizations
A
(1)
and
A
(2)
. Assigning
U
1
:¼ U
2
:¼ e
1
yields a core tensor
C
11
with entries
c
1
;
1
;
1
Þ
¼
1,
c
1
;
1
;
2
Þ
¼
0
:
ð
ð
C
with
The remaining three choices give rise to always the same core tensor
entries
c
1
;
1
;
1
^
Þ
¼
1,
^
c
1
;
1
;
2
Þ
¼
0
:
ð
ð
The respectively induced rank-(1,1,2) approximations to
A
are given by
,
1000
0000
A
ðÞ
11
¼
,
0100
0000
A
ðÞ
12
¼
,
0000
1000
A
ðÞ
21
¼
and
0000
0100
A
ðÞ
22
¼
:
In each case, the approximation error is the same.
■
The following properties of the HOSVD have been worked out.
Theorem 9.1 (Theorem 2 in [DLDMV00])
The core tensor of a
(
rank-
n)
HOSVD
of A
n
satisfies:
∈
R
