 This is indeed the case.As another example, for

the reasonable way of calculating "f"("T") would seem to be

However, this expression is not defined if the inverses on the right-hand side do not exist, that is, if either 2 or 5 are eigenvalues of "T".For a given matrix "T", the eigenvalues of "T" dictate to what extent "f"("T") can be defined; i.e., "f"(λ) must be defined for all eigenvalues λ of "T".