Closed-form expressions for matrix functions (includinge^{At})

In this technical note, we develop closed-form expressions for
f(A)
where
A
is any constant
n \times n
matrix and
f(\lambda)
is any analytic function in a region containing the eigenvalues λ
i
of
A, i = 1,2, ..., n
. The matrix
A
may have nondistinct eigenvalues.