Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n × n matrix 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 conjugate transpose of the vector x.

In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of 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 \{ |\lambda| : \lambda \in W(A) \} = \sup_{\|x\|=1} |\langle Ax, x \rangle|.

r(A) is a norm. r(A)=||A||, where ||A|| is the operator norm of A.

Properties

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

Generalisations

See also

References

    Bibliography
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