The 
-pseudospectrum
of a square matrix 
is the set of eigenvalues that
can result from perturbations of size at most 
in a specified matrix norm.
In the spectral norm, for 
, it is equivalently
 |
where 
is the identity matrix and 
denotes the smallest singular
value. Thus
 |
Here 
may be a complex matrix. These definitions use
the closed-set convention (Pseudospectra Gateway). Pseudospectra quantify the sensitivity
of eigenvalues to perturbations (Trefethen and Embree
2005).
For a normal matrix, this set is the union of closed disks of radius 
centered at the eigenvalues.
For a matrix that is not a normal
matrix, the regions can be much larger. A Grcar matrix
provides a familiar example.
