# Inner product

1. Nov 6, 2006

### devious_

Is there a non-ugly proof of the following identity:
$$\langle Ax,y \rangle = \langle x,A^*y \rangle$$
where A is an nxn matrix over, say, $\mathbb{C}$, A* is its conjugate transpose, and $\langle \cdot , \cdot \rangle$ is the standard inner product on $\mathbb{C} ^n$.

Last edited: Nov 6, 2006
2. Nov 6, 2006

### Galileo

Just start writing out the left side using $$<a,b>=a^\dagger b$$.

3. Nov 6, 2006

### devious_

Thanks.

Although using my definition for the standard inner product I had to use $\langle a , b \rangle = a^t \bar{y}$, but it all worked out in the end.