- Source: Numerical range
In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex
n
×
n
{\displaystyle n\times n}
matrix A is the set
W
(
A
)
=
{
x
∗
A
x
x
∗
x
∣
x
∈
C
n
,
x
≠
0
}
{\displaystyle W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}}
where
x
∗
{\displaystyle \mathbf {x} ^{*}}
denotes the conjugate transpose of the vector
x
{\displaystyle \mathbf {x} }
. The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors).
In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of the numerical range are used to study quantum computing.
A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e.
r
(
A
)
=
sup
{
|
λ
|
:
λ
∈
W
(
A
)
}
=
sup
‖
x
‖
=
1
|
⟨
A
x
,
x
⟩
|
.
{\displaystyle r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|=1}|\langle Ax,x\rangle |.}
Properties
The numerical range is the range of the Rayleigh quotient.
(Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
W
(
α
A
+
β
I
)
=
α
W
(
A
)
+
{
β
}
{\displaystyle W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}}
for all square matrix
A
{\displaystyle A}
and complex numbers
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. Here
I
{\displaystyle I}
is the identity matrix.
W
(
A
)
{\displaystyle W(A)}
is a subset of the closed right half-plane if and only if
A
+
A
∗
{\displaystyle A+A^{*}}
is positive semidefinite.
The numerical range
W
(
⋅
)
{\displaystyle W(\cdot )}
is the only function on the set of square matrices that satisfies (2), (3) and (4).
(Sub-additive)
W
(
A
+
B
)
⊆
W
(
A
)
+
W
(
B
)
{\displaystyle W(A+B)\subseteq W(A)+W(B)}
, where the sum on the right-hand side denotes a sumset.
W
(
A
)
{\displaystyle W(A)}
contains all the eigenvalues of
A
{\displaystyle A}
.
The numerical range of a
2
×
2
{\displaystyle 2\times 2}
matrix is a filled ellipse.
W
(
A
)
{\displaystyle W(A)}
is a real line segment
[
α
,
β
]
{\displaystyle [\alpha ,\beta ]}
if and only if
A
{\displaystyle A}
is a Hermitian matrix with its smallest and the largest eigenvalues being
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
.
If
A
{\displaystyle A}
is a normal matrix then
W
(
A
)
{\displaystyle W(A)}
is the convex hull of its eigenvalues.
If
α
{\displaystyle \alpha }
is a sharp point on the boundary of
W
(
A
)
{\displaystyle W(A)}
, then
α
{\displaystyle \alpha }
is a normal eigenvalue of
A
{\displaystyle A}
.
r
(
⋅
)
{\displaystyle r(\cdot )}
is a norm on the space of
n
×
n
{\displaystyle n\times n}
matrices.
r
(
A
)
≤
‖
A
‖
≤
2
r
(
A
)
{\displaystyle r(A)\leq \|A\|\leq 2r(A)}
, where
‖
⋅
‖
{\displaystyle \|\cdot \|}
denotes the operator norm.
r
(
A
n
)
≤
r
(
A
)
n
{\displaystyle r(A^{n})\leq r(A)^{n}}
Generalisations
C-numerical range
Higher-rank numerical range
Joint numerical range
Product numerical range
Polynomial numerical hull
See also
Spectral theory
Rayleigh quotient
Workshop on Numerical Ranges and Numerical Radii
Bibliography
Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58 (1): 77–91, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8, S2CID 119427312.
Dirr, G.; Helmkel, U.; Kleinsteuber, M.; Schulte-Herbrüggen, Th. (2006), "A new type of C-numerical range arising in quantum computing", Proc. Appl. Math. Mech., 6: 711–712, doi:10.1002/pamm.200610336.
Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press, ISBN 978-0-521-07988-4.
Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges II, Cambridge University Press, ISBN 978-0-521-20227-5.
Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, Chapter 1, ISBN 978-0-521-46713-1.
Horn, Roger A.; Johnson, Charles R. (1990), Matrix Analysis, Cambridge University Press, Ch. 5.7, ex. 21, ISBN 0-521-30586-1
Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc., 124 (7): 1985, doi:10.1090/S0002-9939-96-03307-2.
Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications, 252 (1–3): 115, doi:10.1016/0024-3795(95)00674-5.
"Functional Characterizations of the Field of Values and the Convex Hull of the Spectrum", Charles R. Johnson, Proceedings of the American Mathematical Society, 61(2):201-204, Dec 1976.
References
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- Hujan
- Daftar algoritme
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- Numerical range
- Workshop on Numerical Ranges and Numerical Radii
- Product numerical range
- Numerical analysis
- Numerical aperture
- Numerical differentiation
- Numerical weather prediction
- Numerical Recipes
- List of numerical libraries
- European Centre for Medium-Range Weather Forecasts
