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.


Generally one requires that an inner product on a complex vector space be linear in one argument and conjugate linear in the other. What you have proven is precisely that this inner product is conjugate linear in the first argument.

Note that linearity in the second argument and the conjugate symmetry together imply conjugate linearity in the first argument.

  • $\begingroup$ so is this definition defined in wikipedia en.wikipedia.org/wiki/Inner_product_space#Definition not well defined? $\endgroup$ – kaka Mar 14 '15 at 1:24
  • $\begingroup$ The definition on Wikipedia specifies linearity in the first argument instead of the second. In that case, the conjugate symmetry implies conjugate linearity in the second argument. Whether one wants the first or second argument to be linear is a matter of choice; one just needs to make an arbitrary choice and stick with it. $\endgroup$ – Phillip Andreae Mar 14 '15 at 1:40

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