I have a matrix metric space where the distance between two matrices is given by the Frobenius metric. Distance between matrices $A$ and $B$: $$ \sqrt{Tr \left((A-B)^\dagger(A-B)\right)} $$
How do I describe a hyperplane in this metric space? Ideally I would like to understand the geometry of such a space to be able to play around with it. For example be able to draw parallel planes, etc.
The space $\mathbb{R}^{m\times n}$ with the inner product $$ \langle A,B\rangle_F = \mathrm{Tr}(A^T B) $$ can be regarded as the Euclidean space $\mathbb{R}^{mn}$ with the usual inner product $\mathbf{v}\cdot \mathbf{w}=\mathbf{v}^T \mathbf{w}$. You can just use the identification (isomorphism) $$ \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \mapsto (a_{11}, a_{12}, \ldots,a_{21}, a_{22}, \ldots, a_{mn}). $$ In particular, a hyperplane can be described as $$ Tr(A^T X) =b $$ where $A$ is a constant matrix, $b$ a real constant and $X$ a matrix whose components are variables. The matrix $A$ can be interpreted as a vector normal to the hyperplane. By changing the constant $b$, you will get a plane parallel to the original plane.