The behavior of a normal matrix (e.g., a real symmetric matrix or a complex Hermitian matrix) is governed by its eigenvalues; that is, if A is a normal matrix and f is any analytic function, then the 2-norm of f(A) is the maximum absolute value of f on the spectrum of A. The same holds when A is a normal linear operator, except that now the spectrum may include more than just the eigenvalues. This statement does not hold for nonnormal matrices and linear operators, and there is considerable interest in identifying sets in the complex plane that can be associated with nonnormal operators to provide the sort of information that the spectrum provides in the normal case.
There are many reasons for wanting to do this. It would enable us to ``visualize'' a nonnormal matrix as a set in the complex plane and to reason about its behavior through this set, just as eigenvalues enable us to picture a normal matrix. Among the many applications are the study of stability of differential and difference equations, prediction of cutoff phenomena in Markov chains (such as the fact that it takes 7 riffle shuffles to randomize a deck of cards), and understanding the convergence behavior of iterative methods for solving large nonsymmetric systems of linear equations.
Current work deals with finding sets in the complex plane that can be associated with a given nonnormal matrix to give more information than the spectrum alone can provide. Possibilities include the field of values, the epsilon-pseudospectrum, and the polynomial numerical hull of a given degree. Connections with complex analysis, such as Blaschke products and conformal mapping are also being studied. An extremely interesting conjecture was made recently by Michel Crouzeix (See "Bounds for analytical functions of matrices", Integr. Equ. Oper. Theory 48 (2004), pp. 461-477; "Numerical range and functional calculus in Hilbert space", J. Functional Analysis 244 (2007), pp. 668-690): For any matrix A and any polynomial p, the 2-norm of p(A) is less than or equal to twice the maximum absolute value of p on the field of values of A. A proof or disproof of this conjecture would be of great interest.
For more details, see the references:
Some Extensions of the Crouzeix-Palencia Result
submitted to SIAM Jour. Matrix Anal. Appl., 2017 ,
by T. Caldwell, A. Greenbaum, and K. Li.
Numerical Investigation of Crouzeix's Conjecture
accepted in Lin. Alg. Appl., 2017 ,
by A. Greenbaum and M. Overton.
Near Normal Dilations of Nonnormal Matrices and Linear Operators
SIAM Jour. Matrix Anal. Appl. 37 (2016), pp. 1365-1381. ,
by A. Greenbaum, T. Caldwell, and K. Li.
Variational Analysis of the Crouzeix Ratio
Math Programming (2016). ,
by A. Greenbaum, A. Lewis, and M. Overton.
An Algorithm for finding a 2-Similarity Transformation from a Numerical Contraction
to a Contraction
SIAM Jour. Matrix Anal. Appl. (2015), pp. 1248-1262. ,
by D. Choi and A. Greenbaum.
Roots of Matrices in the Study of GMRES Convergence and Crouzeix's Conjecture
SIAM Jour. Matrix Anal. Appl. (2015), pp. 289-301. ,
by D. Choi and A. Greenbaum.
Crouzeix's Conjecture and Perturbed Jordan Blocks
Lin. Alg. Appl. 436 (2011), pp. 2342--2352 ,
by A. Greenbaum and D. Choi.
Upper and Lower Bounds on Norms of Functions of Matrices
Lin. Alg. Appl. 430 (2009), pp. 52--65 ,
by A. Greenbaum.
Characterizations of the Polynomial Numerical Hull of Degree $k$
Lin. Alg. Appl., 419 (2006), pp. 37-47 ,
by J. Burke and A. Greenbaum.
Some Theoretical Results Derived from Polynomial Numerical Hulls of
Jordan Blocks
Elec. Trans. Num. Anal. 18 (2004), pp. 81-90 ,
by A. Greenbaum.
The Polynomial Numerical Hulls of Jordan Blocks and Related Matrices
Lin. Alg. Appl., 374 (2003), pp. 231-246 ,
by V. Faber, A. Greenbaum, and D. Marshall.
Card Shuffling and the Polynomial Numerical Hull of Degree $k$
SIAM Jour. Sci. Comput. 25 (2004), pp. 408-416 ,
by A. Greenbaum.
Generalizations of the Field of Values Useful in the Study of Polynomial
Functions of a Matrix
Lin. Alg. Appl. 347 (2002), pp. 233-249 ,
by A. Greenbaum.