The numerical range of values is a term from functional analysis and linear algebra .
definition
For a complex Hilbert space with scalar product and a bounded linear operator , the numerical range of is given by
H
{\ displaystyle H}
⟨
⋅
,
⋅
⟩
{\ displaystyle \ langle {\ cdot}, {\ cdot} \ rangle}
T
:
H
→
H
{\ displaystyle T \ colon H \ to H}
T
{\ displaystyle T}
W.
(
T
)
: =
{
⟨
T
x
,
x
⟩
:
‖
x
‖
=
1
}
,
{\ displaystyle W (T): = \ left \ {\ langle Tx, x \ rangle: \ lVert x \ rVert = 1 \ right \},}
where is the norm induced by on .
‖
⋅
‖
{\ displaystyle \ lVert {\ cdot} \ rVert}
⟨
⋅
,
⋅
⟩
{\ displaystyle \ langle {\ cdot}, {\ cdot} \ rangle}
H
{\ displaystyle H}
Analogous to the spectral radius , the numerical radius is defined by .
w
(
T
)
: =
sup
{
|
λ
|
:
λ
∈
W.
(
T
)
}
{\ displaystyle w (T): = \ sup \ {| \ lambda |: \ lambda \ in W (T) \}}
In the special case of complex-valued, square matrices , the definition of the numerical value range is equivalent to
A.
∈
C.
n
×
n
{\ displaystyle A \ in \ mathbb {C} ^ {n \ times n}}
W.
(
A.
)
=
{
x
∗
A.
x
x
∗
x
:
x
∈
C.
n
∖
{
0
}
}
.
{\ displaystyle W (A) = \ left \ {{\ frac {x ^ {*} Ax} {x ^ {*} x}}: x \ in \ mathbb {C} ^ {n} \ setminus \ {0 \} \ right \}.}
W.
(
A.
)
{\ displaystyle W (A)}
is here the image area of the Rayleigh quotient .
properties
The following properties apply to constrained linear operators .
T
:
H
→
H
{\ displaystyle T \ colon H \ to H}
W.
(
T
)
⊆
{
λ
∈
C.
:
|
λ
|
≤
‖
T
‖
}
{\ displaystyle W (T) \ subseteq \ {\ lambda \ in \ mathbb {C}: | \ lambda | \ leq \ lVert T \ rVert \}}
or equivalent to it . Here denotes the operator norm of .
w
(
T
)
≤
‖
T
‖
{\ displaystyle w (T) \ leq \ lVert T \ rVert}
‖
T
‖
{\ displaystyle \ lVert T \ rVert}
T
{\ displaystyle T}
The numerical range of is convex . (Toeplitz-Hausdorff theorem)
T
{\ displaystyle T}
The spectrum is in the final of : . Is finite-dimensional, even applies .
σ
(
T
)
{\ displaystyle \ sigma (T)}
W.
(
T
)
{\ displaystyle W (T)}
σ
(
T
)
⊆
W.
(
T
)
¯
{\ displaystyle \ sigma (T) \ subseteq {\ overline {W (T)}}}
H
{\ displaystyle H}
σ
(
T
)
⊆
W.
(
T
)
{\ displaystyle \ sigma (T) \ subseteq W (T)}
Anything for which is true is an eigenvalue of .
λ
∈
W.
(
T
)
{\ displaystyle \ lambda \ in W (T)}
|
λ
|
=
‖
T
‖
{\ displaystyle | \ lambda | = \ lVert T \ rVert}
T
{\ displaystyle T}
Applications
The right real axis intercept of the numerical value range is the logarithmic norm , for a matrix this is
A.
{\ displaystyle A}
μ
(
A.
)
=
Max
{
x
∗
A.
x
:
x
∗
x
=
1
}
.
{\ displaystyle \ mu (A) = \ max \ left \ {x ^ {*} Ax: \ x ^ {*} x = 1 \ right \}.}
With it a limit for the spectral norm of the matrix exponential can be given, it applies
‖
e
t
A.
‖
2
≤
e
t
μ
(
A.
)
,
t
≥
0.
{\ displaystyle \ | e ^ {tA} \ | _ {2} \ leq e ^ {t \ mu (A)}, \ t \ geq 0.}
Because solves the initial value problem . Then it holds for the Euclidean norm that its derivative satisfies the inequality , from which it follows. This corresponds to the limit for the matrix exponential.
y
(
t
)
=
e
t
A.
y
0
{\ displaystyle y (t) = e ^ {tA} y_ {0}}
y
′
(
t
)
=
A.
y
(
t
)
,
y
(
0
)
=
y
0
{\ displaystyle y '(t) = Ay (t), \, y (0) = y_ {0}}
N
(
t
)
=
‖
y
(
t
)
‖
2
{\ displaystyle N (t) = \ | y (t) \ | ^ {2}}
N
′
(
t
)
=
2
y
(
t
)
∗
y
′
(
t
)
=
2
y
(
t
)
∗
A.
y
(
t
)
≤
2
μ
(
A.
)
y
(
t
)
∗
y
(
t
)
=
μ
(
A.
)
N
(
t
)
{\ displaystyle N '(t) = 2y (t) ^ {*} y' (t) = 2y (t) ^ {*} Ay (t) \ leq 2 \ mu (A) y (t) ^ {* } y (t) = \ mu (A) N (t)}
N
(
t
)
≤
e
2
t
μ
(
A.
)
N
(
0
)
{\ displaystyle N (t) \ leq e ^ {2t \ mu (A)} N (0)}
literature
E. Hairer, G. Wanner, Solving ordinary differential equations II , Springer, 1991.
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