Biomedical Engineering Reference
In-Depth Information
x
T
a
∂
a
T
x
∂
∂
=
∂
=
a
,
(C.82)
x
x
x
T
Ax
∂
∂
A
T
=
(
A
+
)
x
,
(C.83)
x
xa
T
ax
T
∂
tr
(
)
=
∂
tr
(
)
=
a
.
(C.84)
∂
x
∂
x
th element of a matrix
A
as
A
i
,
j
, Differentiating a scalar
F
with a matrix
A
is defined as creating a matrix whose
Let us denote the
(
i
,
j
)
(
i
,
j
)
th element is equal
to
Representative identities are the following, where
x
and
y
are column
vectors and
A
and
B
are matrices.
∂
F
/∂
A
i
,
j
.
∂
(
)
tr
A
=
,
I
(C.85)
∂
A
∂
(
)
tr
AB
B
T
=
,
(C.86)
∂
A
A
T
B
∂
tr
(
)
=
B
,
(C.87)
∂
A
ABB
T
∂
(
)
tr
B
T
=
(
+
),
A
B
(C.88)
∂
A
x
T
Ay
∂
∂
xy
T
=
,
(C.89)
A
∂
log
|
A
|
A
−
1
T
=
(
)
.
(C.90)
∂
A
C.7 Several Formulae for Matrix Computations
Used in this Topic
The following are representative formulae for the matrix inversion.
BD
−
1
C
)
−
1
A
−
1
A
−
1
B
CA
−
1
B
)
−
1
CA
−
1
(
A
+
=
−
(
D
+
,
(C.91)
A
−
1
B
T
C
−
1
B
)
−
1
B
T
C
−
1
AB
T
BAB
T
)
−
1
(
+
=
(
+
C
.
(C.92)
Also, we have
AB
CD
−
1
M
D
−
1
CMBD
−
1
MBD
−
1
−
=
,
(C.93)
D
−
1
CM D
−
1
−
+
where
BD
−
1
C
)
−
1
M
=
(
A
−
.
