Inner product practice problem

Question

This is a practice question for the final exam but I'm not sure how to start it. Is it possible to multiply vector v and w into matrix A?

Answer 1

You want to check the inner product properties, see here Inner Product Definition. Ill show you a few, then you can do the rest. Let $x,y,z\in \mathbb{C}^2$, $\alpha, \beta \in \mathbb{C}$:

(1) Linearity in first slot: $$ \langle x+z , y \rangle = (x+z) A (y)^* = xA(y*) +zAy^* = xAy^* + zAy^* = \langle x, y\rangle + \langle z, y\rangle $$ $$ \langle \alpha x, y\rangle = \alpha x A y^* = \alpha (xAy^*) = \alpha \langle x,y\rangle$$

(2) Conjugate Symmetry: $$\langle x,y \rangle = x Ay^* = (yA^*x^*)^* = (y Ax^*)^* = \langle y ,x\rangle^*$$ Where I have used that the conjugate transpose of $A$ is $A$ itself, which you can check. You got it from here?

Regarding the conjugate issue in the comments: $$A^* = \bar{A}^T = \overline{\begin{pmatrix}3 & i \\ -i & 2 \end{pmatrix}}^T = \begin{pmatrix}3 & -i \\ i & 2 \end{pmatrix}^T = \begin{pmatrix} 3 & i \\ -i & 2 \end{pmatrix} = A$$