In paper it says that the inner product of two $n$-dimensional vectors is equal to:
$$\langle X,Y\rangle :=X^*MY $$
where $M$ is Hermitian and positive definite.
It is easy to proof the conjugate symmetry and positive definiteness but linearity in first argument is bit confusing:
$$\langle \alpha X,Y\rangle =(\alpha X)^*MY= \alpha^* X^*MY \neq \alpha \langle X,Y\rangle .$$
If we assume linearity in the second argument, then it is linear. But according to the given definition of above inner product, it seems to be not linear in first argument.
Any idea please.