# $\langle A, B\rangle := \operatorname{Tr}(AB^{∗})$ Inner Product in $\mathbb{C}$?

We define $$\langle\cdot, \cdot\rangle$$: $$\mathbb{C}^{n\times n} \times \mathbb{C}^{n\times n}\longrightarrow \mathbb{C}$$ by $$\langle A, B\rangle := \operatorname{Tr}(AB^{∗})$$. Show that this mapping gives an inner product on the vector space $$\mathbb{C}^{n\times n}$$.

Does someone have an idea how I can prove that?

• Can you write the definition of an inner product and tell us which property you are unable to verify? Jan 21 '19 at 8:55
• just show that it is positive definite, symmetrical (well in this case conjugate symmetrical) and linear with scalar Jan 21 '19 at 8:57
• Jan 21 '19 at 15:03

An inner product space is a vector space $$\mathbb{C}^{n×n} \times \mathbb{C}^{n×n}$$ over the field $$\mathbb{C}$$ together with an inner product, i.e., with a map $$\mathbb{C}^{n×n} \times \mathbb{C}^{n×n}\longrightarrow \mathbb{C}$$ such that satisties

1. Conjugate symmetry:

$$\langle A,B\rangle ={\overline {\langle B,A\rangle }}$$

2. Linearity in the first argument:

$$\langle aA,B\rangle =a\langle A,B\rangle \\\langle A+B,C\rangle =\langle A,C\rangle +\langle B,C\rangle$$

3. Positive-definiteness:

$$\langle x,x\rangle \geq 0\\\langle x,x\rangle =0\Leftrightarrow x=\mathbf {0} \,.$$

where $$A,B,C \in \mathbb{C}^{n×n} \times \mathbb{C}^{n×n}$$ and $$a\in \mathbb{C}$$.

HINT:

Use the following facts.

$$Tr(aA)=aTr(A)$$ $$Tr(A)=Tr(A)^T$$ $$Tr(AB)=Tr(BA)$$ $$Tr(AB^T)=Tr(A^TB)=Tr(B^TA)=Tr(BA^T)$$

• You're right, I just took the inverse equation and apply $Tr$ both sides. My bad. Jan 21 '19 at 15:49