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I was reading this post, Can you transpose a matrix using matrix multiplication?, and I thought it was interesting that we can't get a "transpose matrix" $B$ such that $BA = A^T$, at least not for all $A$. My questions are:

Given a matrix $A$, what kinds of matrices can or cannot be obtained by left multiplication $BA$? What about right multiplication $AC$? What if we allow both left and right multiplication, like $BAC$?

Does it depend heavily on the matrix $A$, or are there certain types of matrices that can always be obtained or can never be obtained?

I suppose what I am asking is what sorts of linear maps we can get by left composition and right composition, and why certain maps cannot be achieved.

I would also appreciate if someone could let me know what area of math this kind of question belongs to or is related to (e.g. some subfield of abstract algebra?), and how it might be important.

Edit: as pointed out in the comments, there are some familiar examples: For $A = 0$, left or right multiplication by anything still gives $0$. For $A = I$, it is the identity, so any matrix is obtainable. For invertible matrices, we can get the identity by left or right multiplication by its inverse; for non-invertible matrices, this is impossible.

twosigma
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  • The answer indeed depends on $A$. For example, for $A=0$, only the zero matrix can be obtained by left multiplication from it. For $A=I$, all matrices can be obtained. We can solve $AB=C$ if $A$ is invertible, namely $B=A^{-1}C$. So then any $C$ can be obtained from $A$. – Dietrich Burde May 29 '20 at 14:13
  • For square matrices, $A^T$ can always be obtained from $A$ as $A^T=BAC$. Reduce $A$ by Gaussian elimination $EAF = D$, where $D$ consists of $1$s and $0$s. Then $D=D^T=F^TA^TE^T$, so $A^T=F^{-T}EAFE^{-T}$. – Chrystomath May 29 '20 at 15:59

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You are entering into the land of ring theory.

The $n\times n$ matrices over a ring $R$, denoted $M_n(R)$ forms a ring. The set of matrices $\{BA\mid B\in M_N(R)\}$ is what is called a left ideal of $M_n(R)$. In fact, one would call it a principal left ideal.

Looking at what $BA$ means, you can say that the possible results will be every matrix whose rows lie in the row space of $A$.

The set $\{BAC\mid B,C\in M_n(R)\}$ is not very special in itself. If you take the collection of finite sums of elements of this form, you do get something special: a (two-sided) ideal.

rschwieb
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