A matrix function 
extends a scalar function 
to a square complex matrix 
through its Jordan
canonical form (Higham 2008). Suppose 
is analytic on an
open set containing all eigenvalues
of 
. For a Jordan block 
of size 
, where 
is the identity matrix
and 
, define
 |
If 
is a Jordan matrix decomposition, apply
this rule to each Jordan block and set 
. The result is independent of the choice
of Jordan basis. For a diagonalizable
matrix, only the values 
at its eigenvalues
are needed. Nontrivial Jordan blocks also require
derivatives.
Equivalently, the Cauchy integral formula gives
 |
where 
consists of positively oriented, piecewise smooth, simple closed contours
with disjoint interiors enclosing all eigenvalues,
and 
is analytic on and inside each contour
(Higham 2008, 2020). The matrix 
is the resolvent
operator of 
. This construction is a finite-dimensional
functional calculus. For a scalar polynomial,
it agrees with substitution of matrix powers. The
choice 
gives the matrix exponential.
