I found the following question and associated thread interesting, and I’d like to contribute:
Product rule for matrix-valued functions and Differentiability of matrix multiplication.
However, I have not used stackexchange long enough to gain enough “reputation” points to be allowed to comment. Moreover, there are technical errors in answers that were given, and I was willing to just email the authors to let them know, so they can edit their explanations, but I cannot find a way to do that either. What have I misunderstood about the way I, a mathematician who has practiced the craft quite rigorously for over thirty-five years, can contribute to stackexchange discussions about mathematical topics in which my expertise and experience can be beneficial to all concerned?
To be sure, in the post referenced above, there are two answers and both make sense, but the first one has no associated comments, although it begins with a fundamentally flawed description of the domain of the operators that are involved. The problem is about differentiability of a product of differentiable operators on a space of square matrices. The first answer given is erroneous from the beginning because while it properly evaluates the derivative of a given differentiable operator F at a square matrix $M_0$, it then evaluates the result at a column vector $M$. This is technically incorrect, since the tangent space to the domain manifold (the space of square matrices of a fixed given size) is the space itself, up to (topological and algebraic) isomorphism in the category of Banach spaces. That is, instead of evaluating $F^{\prime}(M_0)$ at a column vector $M$, the author of that answer should evaluate it at a square matrix, because technically, no column vector is in the domain of the linear operator $F^{\prime}(M_0)$.
Another reason I ask why I’m not allowed to participate in such discussions is that I would like to inform the original questioner that the title of the question is incorrect, in the sense that it is misleading. The question asked is not at all about “differentiability of matrix multiplication”. A much better title would be “Product rule for matrix-valued functions”, or “Differentiability of products of matrix-valued functions”. Given a space $\mathcal{M}$ of square matrices over the reals of a given size, the binary operation we call “matrix multiplication of matrices in $\mathcal{M}$” is a function from $\mathcal{M}\times\mathcal{M}$ into $\mathcal{M}$, and in the question body,it is clear that this is not what the questioner means to ask about. The questioner should be encouraged to edit the question title accordingly, because the following question is fundamentally different:
“Is matrix multiplication differentiable?”.
The second answer given answers the above question, so it answers what the question title asks, and nicely uses that answer to aid in answering the question asked in the body of the question at hand.
