Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range of a square matrix with complex entries is a subset of the complex plane associated to the matrix. If A is an n × n matrix with complex entries, then the numerical range of A is the set

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

where x* denotes the Hermitian adjoint of the vector x. In other words, it is the range of the Rayleigh quotient. The numerical range is also called the field of values.^[1]

Numerical radius

In a way analogous to spectral radius, the numerical radius of an operator T on a complex Hilbert space, denoted by w(T), is defined by

$w(T) = \sup \{ |\lambda| : \lambda \in W(T) \} = \sup_{\|x\|=1} |\langle Tx, x \rangle|$.

w is then a norm. It is equivalent to the operator norm by the inequality:

$2^{-1} \| T \| \le w(T) \le \| T \|$

Paul Halmos conjectured:

$w(T^n) \le w(T)^n$

for every integer n > 0. It was later confirmed by Charles Berger and Carl Pearcy.^[2]

Some theorems

The Hausdorff–Toeplitz theorem states that the numerical range of any matrix is a convex set.^[3] Furthermore, the spectrum of A is contained within the closure of W(A).^[4] If A is a normal matrix, then the numerical range is the polygon in the complex plane whose vertices are eigenvalues of A.^[5] In particular, if A is Hermitian then the polygon reduces to the segment of the real axis bounded by the smallest and the largest eigenvalue,

$W(A) \ = \ [\lambda_{\rm min}, \ \lambda_{\rm max} ]$

which explains the name numerical range. If A is not normal, then a weaker property holds: any "corner" of the numerical range is an eigenvalue of A. Here, the precise definition of a "corner" is that of a sharp point: a point w on the boundary of a set S ⊂ C is called a sharp point of S if there exist two angles θ₁ and θ₂ with 0 ≤ θ₁ < θ₂ < 2π such that for all z ∈ S and for all θ ∈ (θ₁, θ₂) the inequality Re e^iθw ≥ Re e^iθz holds.^[6]

Bounded operators on a Hilbert space

If the closure of the numerical range of a bounded operator coincides with the convex hull of its spectrum, then it is called a convexoid operator. An example of such an operator is a normal operator.

Special cases

• matrices of order N=2. Numerical range of any operator A of order two forms an elliptical disk in the complex plane with both eigenvalues $\lambda_{1}, \ \lambda_{2}$ as foci and minor axis equal to $\sqrt{ {\rm Tr} (A^* A) - |\lambda_{1}|^2 - |\lambda_{2}|^2 }$. In the special case of a normal A the disk reduces to an interval, $[\lambda_{1}, \ \lambda_{2} ]$ - see Li 1996.

• matrices of order N=3. Numerical range forms a) an ellipse; b) has an ovular shape; c) has a flat portion of its boundary; d) is a convex hull of a point and an ellipse e) is a triangle (for normal operators only) - see Keeler, Rodman and Spitkovsky 1997.

Generalisations

• C - numerical range
• Higher rank numerical range
• Joint numerical range
• Product numerical range

See also

• Field of values
• Spectral theory

Notes

1. ^ Horn & Johnson (1991, Definition 1.1.1)
2. ^ Lax, Linear algebra and its applications, 2nd ed.
3. ^ Horn & Johnson (1991, Property 1.2.2)
4. ^ Horn & Johnson (1991, Property 1.2.6)
5. ^ Horn & Johnson (1991, Property 1.2.9)
6. ^ Horn & Johnson (1991, Theorem 1.6.3)

References

• Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1 .

• Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc. 124, 1985 .

• Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of $3 \times 3$ matrices", Linear Algebra Applications 252, 115 .