Properties
- The numerical range is the range of the Rayleigh quotient.
- (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
- for all square matrix A and complex numbers α and β. Here I is the identity matrix.
- is a subset of the closed right half-plane if and only if is positive semidefinite.
- The numerical range is the only function on the set of square matrices that satisfies (2), (3) and (4).
- (Sub-additive) .
- contains all the eigenvalues of A.
- The numerical range of a 2×2 matrix is an elliptical disk.
- is a real line segment if and only if A is a Hermitian matrix with its smallest and the largest eigenvalues being α and β
- If A is a normal matrix then is the convex hull of its eigenvalues.
- If α is a sharp point on the boundary of, then α is a normal eigenvalue of A.
- is a norm on the space of n×n matrices.
Read more about this topic: Numerical Range
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)