Frobenius Inner Product

Last updated Nov 1, 2022

Let $n, m \in \mathbb{N}$. Then $$\langle A, B \rangle := \text{tr} (A^{T} B)$$. This defines an Inner Product on $\mathbb{R}^{n \times m}$.

We need to establish that we indeed have an Inner Product. We check the conditions: TODO

1. Observe that $\langle A, B \rangle = \text{tr} (A^{T} B) = \sum\limits_{j=1}^{m} A_{\cdot j} \cdot B_{\cdot j} = \sum\limits_{j=1}^{m} \sum\limits_{i=1}^{n} A_{ij} B_{ij}$, where $A_{\cdot j}$ denotes the $j$th column of $A$. This establishes that this Inner Product is just the Dot Product when $A$ and $B$ are rolled out into vectors of $\mathbb{R}^{nm}$.

• Euclidean Space
• Natural Numbers
• Real Matrix Vector Space