Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex ${\displaystyle n\times n}$ matrix A is the set

${\displaystyle W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}}$

where ${\displaystyle \mathbf {x} ^{*}}$ denotes the conjugate transpose of the vector ${\displaystyle \mathbf {x} }$. 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.

${\displaystyle r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|=1}|\langle Ax,x\rangle |.}$

Properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. ${\displaystyle W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}}$ for all square matrix ${\displaystyle A}$ and complex numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. Here ${\displaystyle I}$ is the identity matrix.
4. ${\displaystyle W(A)}$ is a subset of the closed right half-plane if and only if ${\displaystyle A+A^{*}}$ is positive semidefinite.
5. The numerical range ${\displaystyle W(\cdot )}$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. (Sub-additive) ${\displaystyle W(A+B)\subseteq W(A)+W(B)}$, where the sum on the right-hand side denotes a sumset.
7. ${\displaystyle W(A)}$ contains all the eigenvalues of ${\displaystyle A}$.
8. The numerical range of a ${\displaystyle 2\times 2}$ matrix is a filled ellipse.
9. ${\displaystyle W(A)}$ is a real line segment ${\displaystyle [\alpha ,\beta ]}$ if and only if ${\displaystyle A}$ is a Hermitian matrix with its smallest and the largest eigenvalues being ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.
10. If ${\displaystyle A}$ is a normal matrix then ${\displaystyle W(A)}$ is the convex hull of its eigenvalues.
11. If ${\displaystyle \alpha }$ is a sharp point on the boundary of ${\displaystyle W(A)}$, then ${\displaystyle \alpha }$ is a normal eigenvalue of ${\displaystyle A}$.
12. ${\displaystyle r(\cdot )}$ is a norm on the space of ${\displaystyle n\times n}$ matrices.
13. ${\displaystyle r(A)\leq \|A\|\leq 2r(A)}$, where ${\displaystyle \|\cdot \|}$ denotes the operator norm.^[1][2][3][4]
14. ${\displaystyle r(A^{n})\leq r(A)^{n}}$

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

1. ""well-known" inequality for numerical radius of an operator". StackExchange.
2. "Upper bound for norm of Hilbert space operator". StackExchange.
3. "Inequalities for numerical radius of complex Hilbert space operator". StackExchange.
4. Hilary Priestley. "B4b hilbert spaces: extended synopses 9. Spectral theory" (PDF). In fact, ‖T‖ = max(−m_T , M_T) = w_T. This fails for non-self-adjoint operators, but w_T ≤ ‖T‖ ≤ 2w_T in the complex case.