Inner Products on the Space of Complex Square Matrices

In this paper we study the problem of finding explicit expressions for inner products on the space of complex square matrices $\Mn$. We show that, given an inner product $\lip \cdot, \cdot \rip$ on $\Mn$, with some conditions, there exist positive matrices $A_j$ and $B_j \in \Mn$, for $j=1, 2\dots, m$ such that $$\lip X, Y \rip = \sum_{j=1}^m \tr\left(Y^* A_j X B_j \right),$$ for all $X, Y \in \Mn$. However, we show that the result does not hold for all inner products. In fact, if the above expression does not hold, we show that there exist positive matrices $A_j$ and $B_j \in \Mn$, for $j=1, 2\dots, m$ such that $$\lip X, Y \rip = -\tr\left(Y^* A_1 X B_1 \right)+ \sum_{j=2}^m \tr\left(Y^* A_j X B_j \right),$$ for all $X, Y \in \Mn$.