Numerical Range

Introduction:-

The numerical range of an n x n complex matrix Q, also known as its field of values, is defined as


where z* denotes the complex conjugate transpose of z. Following diagram illustrates the numerical range as an image of the unit sphere.
 

Basic Properties:-

Many useful properties are associated with numerical range. Some of the common properties of numerical range are listed below:
  1. F(Q) is closed and bounded.
  2. F(Q) is convex
  3. F(U*QU) = F(Q) for any n x n complex unitary matrix U.
  4. F(Q) is closed interval of real line if Q is Hermitian.
 

Picture Gallery:-

Classical Numerical Range
    Orthogonal projection of the numerical range
    Critical points and sharp points of the template
    Generic and normal properties of complex matrices
 
    Examples
        Generic matrices
           2 x 2 matrix
           3 x 3 matrix
           3 x 3 matrix another example
           4 x 4 matrix
           5 x 5 matrix
           6 x 6 matrix
        Nongeneric matrices
           2 x 2 matrix
           3 x 3 matrix
           5 x 5 matrix
Generalized Numerical Range Examples

       2 Block Cases [Critical value curves]

            3 x 3 doubly stochastic matrix: Template and critical eigenvalues

         3 x 3 generic matrix: Template for non-zero alpha

          3 x 3 generic matrix: Critical Eigenvalues for non-zero alpha

          4 x 4 genreic matrix: Template and critical eigenvalues

          4 x 4 generic matrix: critical eigenvalues for varying alpha
 
 

        3 Block Cases [Critical Value Surfaces]

          3 x 3 matrix with three blocks

            Gallery is still under construction
 
 
 
 

 Control Systems
    Perturbed multivariable feedback system
    A closed loop feedback system
 

 
 

Selected Publications:-

Following papers explore these properties in detail. Particularly, the convexity of the template is of special interest in control applications. The following papers explore  results therein are based on convexity of the template.

Journal Papers

 
  1. Edmond A. Jonckheere, Farooq Ahmad, and Eugene Gutkin, "Differential Topology of Numerical Range," Linear Algebra and its Applications, vol. 279, pp. 227-254, 1998.
  2. Eugene Gutkin, Edmond A. Jonckheere, and Michael Karow, "Convexity of the joint numerical range: topological and differential geometric viewpoints," Linear Algebra and its Applications, vol. 376, pp. 143-171, 2003.


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