Numerical range explained

n x n matrix A is the setW(A) = \left\{ (x ^{*Ax})/(x ^{*x}) \mid x \in C ^{n, x\not=0\right\}}

where

x ^{*} denotes the conjugate transpose of the vector 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 \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.

W(\alpha A + \beta I) = \alpha W(A) + \{\beta\} for all square matrix A and complex numbers \alpha and \beta . Here I is the identity matrix.W(A) is a subset of the closed right half-plane if and only if A + A ^{*} is positive semidefinite.

1. The numerical range

W( ⋅ ) is the only function on the set of square matrices that satisfies (2), (3) and (4).

1. (Sub-additive)

W(A + B)\subseteq W(A) + W(B) , where the sum on the right-hand side denotes a sumset.W(A) contains all the eigenvalues of A .

1. The numerical range of a

2 x 2 matrix is a filled ellipse.W(A) is a real line segment [\alpha, \beta] if and only if A is a Hermitian matrix with its smallest and the largest eigenvalues being \alpha and \beta .

1. If

A is a normal matrix then W(A) is the convex hull of its eigenvalues.

1. If

\alpha is a sharp point on the boundary of W(A) , then \alpha is a normal eigenvalue of A .r( ⋅ ) is a norm on the space of n x n matrices.r(A) \leq \|A\| \leq 2 r(A) , where \| ⋅ \| denotes the operator norm.^{[1] [2] [3] [4]} r(A ^{n)} \le 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

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• "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.

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..