# Linearity in first argument of $\langle X,Y\rangle =X^*MY$

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.