Biomedical Engineering Reference
In-Depth Information
2.3.2 Eigenstructure and characteristic fields
The eigenstructure of the first-order system (2.12), (2.13) is that of the principal part
of the system and is given by the eigenvalues and corresponding eigenvectors.
Proposition 3.1.
The eigenvalues of (2.12) are all real and given by
λ
=
u
−
c
,
λ
=
0
,
λ
=
u
+
c
,
(2.14)
1
2
3
A
ρ
ψ
A
where
c
=
(2.15)
is the
wave speed
, analogous to the
sound speed in gas dynamics
[22].
Proof.
By definition the eigenvalues of system (2.12), (2.13) are the eigenvalues of
the matrix
A
, which in turn are the roots of the characteristic polynomial
(
λ
)=
(
−
λ
)=
,
P
Det
A
I
0
(2.16)
where
I
is the identity matrix and
λ
is a parameter. Simple calculations give
(
λ
)=
λ
λ
c
2
=
2
u
2
P
−
2
u
λ
+
−
0
,
from which the result (2.14) follows.
Proposition 3.2.
The right eigenvectors of
A
corresponding to the eigenvalues
(2.14) are
⎡
⎤
⎡
⎤
⎡
⎤
c
2
1
1
⎣
⎦
,
u
2
−
c
2
⎣
⎦
,
⎣
⎦
,
R
1
=
γ
1
u
−
c
R
2
=
γ
2
R
3
=
γ
3
u
+
c
(2.17)
0
1
0
0
where
γ
1
,
γ
2
and
γ
3
are arbitrary scaling factors.
T
we have
=[
,
,
]
Proof.
For an arbitrary right eigenvector
R
r
1
r
2
r
3
AR
=
λ
R
,
(2.18)
which gives the algebraic system
⎫
⎬
r
2
=
λ
r
1
,
u
2
c
2
A
(
−
)
r
1
+
2
ur
2
+
ρ
ψ
K
r
3
=
λ
r
2
,
(2.19)
⎭
0
=
λ
r
3
.
By substituting
in (2.19) by the appropriate eigenvalues in (2.14) in turn we arrive
at the sought result.
λ
