Numerical range explained

matrix A is the set

where

denotes the conjugate transpose of the vector . 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).

Equivalently, the elements of $W(A)$ are of the form $\backslash operatorname(AP)$, where $P$ is a Hermitian projection operator from $\backslash C^2$ to a one-dimensional subspace.

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.

Properties

Let sum of sets denote a sumset.

General properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.

for all square matrix and complex numbers and . Here is the identity matrix. is a subset of the closed right half-plane if and only if is positive semidefinite.

1. The numerical range

is the only function on the set of square matrices that satisfies (2), (3) and (4). for any unitary . .

1. If

is Hermitian, then is on the real line. If is anti-Hermitian, then is on the imaginary line. if and only if .

1. (Sub-additive)

. contains all the eigenvalues of .

1. The numerical range of a

matrix is a filled ellipse. is a real line segment if and only if is a Hermitian matrix with its smallest and the largest eigenvalues being and .Normal matrices

1. If $A$ is normal, and $x\; \backslash in\; \backslash operatorname(v\_1,\; \backslash dots,\; v\_k)$, where $v\_1,\; \backslash ldots,\; v\_k$ are eigenvectors of $A$ corresponding to $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_k$, respectively, then $\backslash langle\; x,Ax\backslash rangle\; \backslash in\; \backslash operatorname\backslash left(\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_k\backslash right)$.
2. If

is a normal matrix then is the convex hull of its eigenvalues.

1. If

is a sharp point on the boundary of , then is a normal eigenvalue of .Numerical radius is a unitarily invariant norm on the space of matrices. } \leq 2r(A) , where } denotes the operator norm.^{[1] [2] [3] [4]} } if (but not only if) is normal. .

Proofs

Most of the claims are obvious. Some are not.

General properties

The following proof is due to^[5]

Numerical radius

Generalisations

• C-numerical range
• Joint numerical range
• Product numerical range
• Polynomial numerical hull

Higher-rank numerical range

The numerical range is equivalent to the following definition:$$W(A)\; =\; \backslash$$This allows a generalization to higher-rank numerical ranges, one for each

:^[6] $$W\_k(A)\; =\; \backslash$$ is always closed and convex,^{[7] [8]} but it might be empty. It is guaranteed to be nonempty if , and there exists some such that is empty if .^[9]

See also

• Spectral theory
• Rayleigh quotient
• Workshop on Numerical Ranges and Numerical Radii

Bibliography

Books

• • • .
  • Book: Bhatia, Rajendra . Matrix analysis . 1997 . Springer . 978-0-387-94846-1 . Graduate texts in mathematics . New York Berlin Heidelberg.
• Book: Gustafson . Karl E. . Numerical Range: The Field of Values of Linear Operators and Matrices . Rao . Duggirala K. M. . 1997 . Springer . 978-0-387-94835-5 . Universitext . New York, NY . 10.1007/978-1-4613-8498-4 . 0172-5939.

Papers

• Toeplitz . Otto . 1918 . Das algebraische Analogon zu einem Satze von Fejér . Mathematische Zeitschrift . de . 2 . 1-2 . 187–197 . 10.1007/BF01212904 . 0025-5874 . free.
• Hausdorff . Felix . 1919 . Der Wertvorrat einer Bilinearform . Mathematische Zeitschrift . de . 3 . 1 . 314–316 . 10.1007/BF01292610 . 0025-5874.
• .
  • .
• .
• Johnson . Charles R. . 1976 . Functional characterizations of the field of values and the convex hull of the spectrum . Proceedings of the American Mathematical Society . American Mathematical Society (AMS) . 61 . 2 . 201–204 . 10.1090/s0002-9939-1976-0437555-3 . 0002-9939 . free.

Notes and References

1. Web site: "well-known" inequality for numerical radius of an operator . StackExchange.
2. Web site: Upper bound for norm of Hilbert space operator . StackExchange.
3. Web site: Inequalities for numerical radius of complex Hilbert space operator . StackExchange.
4. Web site: B4b hilbert spaces: extended synopses 9. Spectral theory. Hilary Priestley. Hilary Priestley. 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..
5. Davis . Chandler . June 1971 . The Toeplitz-Hausdorff Theorem Explained . Canadian Mathematical Bulletin . en . 14 . 2 . 245–246 . 10.4153/CMB-1971-042-7 . 0008-4395.
6. Choi . Man-Duen . Kribs . David W. . Życzkowski . Karol . October 2006 . Higher-rank numerical ranges and compression problems . Linear Algebra and its Applications . en . 418 . 2-3 . 828–839 . 10.1016/j.laa.2006.03.019. subscription .
7. Li . Chi-Kwong . Sze . Nung-Sing . 2008 . Canonical Forms, Higher Rank Numerical Ranges, Totally Isotropic Subspaces, and Matrix Equations . Proceedings of the American Mathematical Society . 136 . 9 . 3013–3023 . 0002-9939.
8. Woerdeman . Hugo J. . 2008-01-01 . The higher rank numerical range is convex . Linear and Multilinear Algebra . 56 . 1-2 . 65–67 . 10.1080/03081080701352211 . 0308-1087. subscription .
9. Li . Chi-Kwong . Poon . Yiu-Tung . Sze . Nung-Sing . 2009-06-01 . Condition for the higher rank numerical range to be non-empty . Linear and Multilinear Algebra . 57 . 4 . 365–368 . 10.1080/03081080701786384 . 0308-1087. 0706.1540 .