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.

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.

Some theorems

The Hausdorff–Toeplitz theorem states that the numerical range of any matrix is a convex set. Furthermore, the spectrum of A is contained within the closure of W(A). If A is a normal matrix, then the numerical range is the polygon in the complex plane whose vertices are eigenvalues of A. 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.

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

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 .