Numerical range

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 ( α A + β I ) = α W ( A ) + { β } 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 ( ⋅ ) is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. (Sub-additive) W ( A + B ) ⊆ 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 ( ⋅ ) is a norm on the space of n×n matrices.
13. r ( A n ) ≤ r ( A ) n

Generalisations