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I am looking for the earliest published source that gives and perhaps proves the identity $\det(I+AB)=\det(I+BA)$ where $A$ and $B$ are just rectangular matrices of finite dimensions (as opposed to possibly infinite-dimensional operators of 'trace class', etc.).

The Wikipedia article for this identity attributed it to J. J. Sylvester without citation of any original work of Sylvester, until June 2019 when the page was edited and now attributes this identity to Nathan Aronszajn and Alexander Weinstein. A few months later, Terence Tao edited a 2013 blog post changing the attribution from Sylvester to Aronszajn–Weinstein, presumably following the Wikipedia article.

A. Chervov, G. Falqui and V. Rubtsov. Algebraic properties of Manin matrices 1. (2009), Remark 23 tells me that the name "Aronszajn–Weinstein formula" comes from Tosio Kato. Perturbation Theory for Linear Operators (1966), chapter 4 section 6. Browsing through Kato, I found lines like the following:

For any $\zeta\in P(T)$, $A(T-\zeta)^{-1}$ is then a degenerate operator belonging to ${\scr B}({\sf X})$ and \begin{align*} \omega(\zeta) &=\omega(\zeta;T,A) =\det(1+A(T-\zeta)^{-1})\\ &=\det[(T+A-\zeta)(T-\zeta)^{-1}] \end{align*} is defined...

It definitely seems to have some resemblances to the identity I am pursuing, but I couldn't find the actual identity in any form in his book. This may very well be due to my lack of any knowledge of the topic; perhaps it is simple for the expert to extract from this abstraction the identity in question. For what it's worth, casually browsing the references Kato provides to the original papers (c. 1940s) of Aronszajn and Weinstein also did not help me. Also, this whole Aronszajn–Weinstein stuff appears very general to me compared to the case for $m\times n$ and $n\times m$ matrices, with $m$ and $n$ finite — surely even if Aronszajn and Weinstein did prove the identity in their general case, someone should have proven the simpler version before them?

An earlier edit of the Wikipedia page for this identity stated without any explicit citation that Sylvester had stated the result without proof in a paper of his of 1857. That said, I've looked through volume 2 of The Collected Mathematical Papers of James Joseph Sylvester, which is supposed to contain all his papers from 1857, and did not find anything useful. It's worth noting however that there's an identity by the name "Sylvester's determinant identity" that stems from J. J. Sylvester. On the relation between the minor determinants of linearly equivalent quadratic functions (1851).

In summary, I would like to see a paper of J. J. Sylvester, or perhaps someone earlier, containing the identity $\det(I+AB)=\det(I+BA)$. It would be nice if the paper contained a proof, but it isn't necessary. As an aside, though not as important, it would also be nice to see where the first usage of the term "Weinstein–Aronszajn identity" as referring to just the identity $\det(I+AB)=\det(I+BA)$ showed up, preferably from before 2009.

Thanks.

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    $\begingroup$ The page you cite refers you to Akritas et al 1996 attributes it to S's 1951 paper. $\endgroup$ – Cosmas Zachos Jan 18 at 23:18
  • $\begingroup$ See equation above (4) in Akritas et al, with W=Z=I. $\endgroup$ – Cosmas Zachos Jan 18 at 23:24
  • $\begingroup$ This is the other identity of Sylvester that I mentioned in the second last paragraph, which is not the one I am looking for. $\endgroup$ – ho boon suan Jan 19 at 2:01
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    $\begingroup$ There is no "other identity"! Interchange the roles of W and Z, as in the WP proof. Do you appreciate the logic? That is why Sylvester did not bother with details. $\endgroup$ – Cosmas Zachos Jan 19 at 14:31
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    $\begingroup$ Perhaps, you should post an answer. Wikipedia clearly treats these identities as distinct, and truth be told, it is hard to see someone coming up with the OP identity from looking at Sylvester's. I suspect it never occurred to him. So spelling out the interrelations in detail would be helpful. Btw, what is "WP proof"? $\endgroup$ – Conifold Jan 21 at 5:41

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