Real $(n,n)$ (square) matrices are obviously a vector space. So we can consider linear transformations form this vector space to itself. Such a transformation $\mathsf{L}$ maps an $(n,n)$ matrix $\pmb{X}$ into another $(n,n)$ matrix, that is, $\mathsf{L}(\pmb{X})$ is an $(n,n)$ matrix and we also have $\mathsf{L}(a\pmb{X}+b\pmb{Y}) = a\, \mathsf{L}(\pmb{X}) + b\,\mathsf{L}(\pmb{Y})$ for every pair of $(n,n)$ matrices $\pmb{X}$ and $\pmb{Y}$, and every pair of real numbers $a$ and $b$.
What is the most general form of such a linear transformation, represented in terms of matrix multiplication and addition? I guess it must have the form $$ \mathsf{L} \colon \pmb{X} \mapsto \sum_{i=1}^{k} \pmb{A}_i\,\pmb{X}\,\pmb{B}_i $$ for some $(n,n)$ matrices $\pmb{A}_1, \dotsc, \pmb{A}_k$ and $\pmb{B}_1, \dotsc, \pmb{B}_k$.
- Is this correct?
If so, then:
Are there general theorems that allow $\pmb{A}_i$ and $\pmb{B}_i$ to have specific properties a priori (eg, can they always be symmetric?); or that set a minimum value of $k$?
How are properties of the linear operator $\mathsf{L}$ – eg rank, determinant, symmetry or antisymmetry, eigensystem, transpose, inverse, and so on – reflected in the properties of $\pmb{A}_i$, $\pmb{B}_i$, $k$?
How does this representation and its properties generalize to affine transformations?
What are good references where to study this representation?
Thanks a bunch!
Any linear map can be represented this way. Consider a matrix $A$ that has $1$ on position $i_1,j_1$ and zeros elsewhere, and matrix $B$ that has value $x$ on position $i_2,j_2$ and zero elsewhere. So then $AXB$ has value $x X_{j_1, \,i_2}$ on position $(i_2, j_1)$. So you can represent any linear map from matrices to matrices in this form -- at least if $k=n^4$.
If $A$ has shape $(m, n)$, it's still the same --- just $A_i$'s would have shape $(m,m)$ and $B_j$'s would have shape $(n,n)$.