  In quantum theory, a physical observable is represented by a Hermitian
operator as it admits real eigenvalues. This stems from the fact that any
measuring apparatus that is supposed to measure a physical observable will
always yield a real number. However, reality of eigenvalue of some operator
does not mean that it is necessarily Hermitian. There are examples of
non-Hermitian operators which may admit real eigenvalues under some symmetry
conditions. However, in general, given a non-Hermitian operator, its average
value in a quantum state is a complex number and there are only very limited
methods available to measure it. Following standard quantum mechanics, we
provide an experimentally feasible protocol to measure the expectation value of
any non-Hermitian operator via weak measurements. The average of a
non-Hermitian operator in a pure state is a complex multiple of the weak value
of the positive semi-definite part of the non-Hermitian operator. We also prove
a new uncertainty relation for any two non-Hermitian operators and show that
the fidelity of a quantum state under quantum channel can be measured using the
average of the corresponding Kraus operators. The importance of our method is
shown in testing the stronger uncertainty relation, verifying the Ramanujan
formula and in measuring the product of non commuting projectors.
