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that can be recalled from an associative memory are more than the ones associated
with the initial matrix W . For an associative memory of N neurons, the possible
patterns become N 2
N
.
10.2.2
Evolution Between the Eigenvector Spaces via Unitary
Rotations
It will be shown that the transition between the vector spaces which are associated
with matrices W
i
's is described by unitary rotations. This is stated in the following
theorem [
165
]:
Theorem 2.
The rotations between the spaces which are spanned by the eigenvec-
tors of the weight matrices
W
i
are unitary operators.
Proof.
Let x
i
;y
i
;
z
i
and x
j
;y
j
;
z
j
be the unit vectors of the bases which span
the spaces associated with the matrices W
i
and W
j
, respectively. Then a memory
vector p can be described in both spaces as: p D .p
x
i
;p
y
i
;p
z
i
/
T
and p D
.p
x
j
;p
y
j
;p
z
j
/
T
. Transition from the reference system W
i
!fx
i
;y
i
;
z
i
g to the
reference system W
j
!fx
j
;y
j
;
z
j
g is expressed by the rotation matrix R, i.e.
p
W
i
D Rp
W
j
. The inverse transition is expressed by the rotation matrix Q, i.e.
p
W
j
D Qp
W
i
.
Furthermore it is true that
0
1
0
1
0
1
p
x
i
p
y
i
p
z
i
x
i
x
j
x
i
y
j
x
i
z
j
y
i
x
j
y
i
y
j
y
i
z
j
z
i
x
j
z
i
y
j
z
i
z
j
p
x
j
p
y
j
p
z
j
@
A
D
@
A
@
A
(10.9)
Thus, using Eq. (
10.9
) the rotation matrices R and Q are given by
0
1
0
1
x
i
x
j
x
i
y
j
x
i
z
j
y
i
x
j
y
i
y
j
y
i
z
j
z
i
x
j
z
i
y
j
z
i
z
j
x
j
x
i
x
j
y
i
x
j
z
i
y
j
x
i
y
j
y
i
y
j
z
i
z
j
x
i
z
j
y
i
z
j
z
i
@
A
;QD
@
A
R D
(10.10)
It holds that
p
W
i
D Rp
W
j
and p
W
j
D Qp
W
i
)
p
W
i
D RQp
W
i
and p
W
j
D QRp
W
j
)
QR D RQ D I)Q D R
1
(10.11)
Moreover, since “dot products” are commutative, from Eq. (
10.10
) one obtains
Q D R
T
. Therefore it holds Q D R
1
D R
T
, and the transition from the reference
W
i
to the reference system
W
j
is described by unitary operators, i.e. QR D
system
R
T
R D R
1
R D I.
