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In this paper we study the problem of finding explicit expressions for innerproducts on the space of complex square matrices $\Mn$. We show that, given aninner product $\lip \cdot, \cdot \rip$ on $\Mn$, with some conditions, thereexist 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 innerproducts. In fact, if the above expression does not hold, we show that thereexist 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 $X, Y \in \Mn$.

Inner Products on the Space of Complex Square Matrices