# What is the most general form of a linear transformation on matrices, written in terms of matrix multiplication and addition? What are its properties?

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!

• matrix multiplication is a "bilinear map" from $M\times M$ to $M$, not a linear one. Most general bilinear map is $\sum_{i,j }\alpha_{ij} A_{ij} B_{ij}$. Otherwise, it's not fully clear what you mean. Oct 15, 2020 at 7:54
• @PeterFranek I rephrased title and question. Is it clearer now? Oct 15, 2020 at 7:56
• I agree that the formula defines a linear map. I don't see what "such a linear operator, represented in terms of matrix multiplication? " means, so first part of the question (if it is correct) is not clear to me. For non-square matrices, you can do the same formula, there is no difference, just $A_i$ and $B_i$ will have other dimension. Oct 15, 2020 at 8:02
• @PeterFranek I tried another rephrasing Oct 15, 2020 at 8:03

## 1 Answer

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 $$(n,n)$$.

• Thank you. I suppose you mean $x\, X_{\dotso,\dotso}$ instead of $xA$. I wonder, though, if it's always possible to use some alternative representation, for example one that uses less than $n^4$ summands, or that uses symmetric matrices, and so on. Oct 15, 2020 at 8:21
• Thank you again. I hope someone can provide at least some literature. I'm not sure about what you say on symmetry: even if $A,B,X$ are symmetric, the product $AXB$ doesn't need to be, unless $A=B$. Oct 15, 2020 at 8:36