Biomedical Engineering Reference
In-Depth Information
=
ʺ
By setting these derivatives to zero, we can derive the relationships,
Ax
Bx
x
T
Ax
. Therefore, the maximum value of
x
T
Ax
is equal to the maximum
eigenvalue of
Ax
ʺ
=
and
Bx
, and the
x
that attains this maximum value is equal to the
eigenvector corresponding to this maximum eigenvalue. Namely, we have
=
ʺ
x
T
Ax
x
T
Bx
=
S
max
{
B
−
1
A
max
x
A
,
B
}=
S
max
{
}
,
and
x
T
Ax
x
T
Bx
=
ˑ
max
{
B
−
1
A
argmax
x
A
,
B
}=
ˑ
max
{
}
.
Using exactly the same derivation, it is easy to show that
x
T
Ax
x
T
Bx
=
S
min
{
min
x
A
,
B
}
,
(C.102)
and
x
T
Ax
x
T
Bx
=
ˑ
min
{
argmin
x
A
,
B
}
.
(C.103)
References
1. F. D. Neeser and J. L. Massey, “Proper complex random processes with applica-
tions to information theory,”
IEEE Transactions on Information Theory
, vol. 39,
pp. 1293-1302, 1993.
2. F. R. Gantmacher,
The Theory of Matrices
. New York, NY: Chelsea Publishing
Company, 1960.