Nonnormal Matrices and Linear Operators, Connections with Complex
Approximation Theory

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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:

* Upper and Lower Bounds on Norms of Functions of Matrices*

** to appear in Lin. Alg. Appl. (see PREPRINTS) **,

by Anne 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.