Inner Products on the Space of Complex Square Matrices

Abstract

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 $X, Y \in \Mn$.