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Therefore, by combining Eqs. (3.21) and (3.22), it follows that:
T
G
×
C
×
A
= D
.
(3.23)
We show that regressor coefficients of
can be obtained by a least square estimation
deduced by Eq. (3.23), by using the
direct product
C
between matrices, also called
the Kronecker product, a special case of
tensor product
used in linear algebra and
in mathematical physics.
Given two real matrices
A
⊗
,
B
of dimension
n
×
m
and
t
×
d
respectively, the
direct
product
:
A
⊗
B
is the matrix of dimension
nt
×
md
, constituted by
nm
blocks
(
A
⊗
B
)
i
,
j
, such that,
if
A
=(
a
i
,
j
|
1
≤
i
≤
n
,
1
≤
j
≤
m
)
,then
(
A
⊗
B
)
i
,
j
=
a
i
,
j
B
(all the elements of
B
are multiplied by
a
i
,
j
). If we use the block notation,
A
⊗
B
can be represented in the
following way:
⎛
⎝
⎞
⎠
.
a
1
,
1
Ba
1
,
11
B
...
a
1
,
m
B
a
2
,
1
Ba
2
,
2
B
a
2
,
mn
B
... ... ... ...
a
n
,
1
Ba
n
,
2
B
...
(3.24)
...
a
n
,
m
B
The Kronecker product is bilinear and associative, that is, it satisfies the following
equations:
A
⊗
(
B
+
C
)=(
A
⊗
B
)+(
A
⊗
C
)
(
A
+
B
)
⊗
C
=(
A
⊗
B
)+(
A
⊗
C
)
(
kA
)
⊗
B
=
A
⊗
(
kB
)=
k
(
A
⊗
B
)
(
A
⊗
B
)
⊗
C
=
A
⊗
(
B
⊗
C
)
.
Moreover, matrix direct product verifies the following equations (the last equation
when matrices are invertible):
(
A
⊗
B
)
×
(
C
⊗
D
)=(
A
×
C
)
⊗
(
B
×
D
)
T
A
T
B
T
(
A
⊗
B
)
=
⊗
)
−
1
A
−
1
B
−
1
(
A
⊗
B
=
⊗
.
The following Lemma asserts a useful property of the Kronecker product [87].
Lemma 3.2 (Vectorization Lemma).
Let us denote by vec
(
W
)
the
vectorization
of
amatrixW
,
obtained by concatenating all the columns of W
,
in their order, in a
unique column vector. Then the following equation holds:
B
T
vec
(
A
×
X
×
B
)=(
⊗
A
)
×
vec
(
X
)
.
(3.25)
