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Assume $A_{N\times{N}}$ is a square matrix. Is there a closed form for $$I+A+AA'+(AA')A'+((AA')A')A'+...$$ Essentially a closed form for multiplication of the matrix by its transpose for infinite times. If yes, then are there any conditions that need to be satisfied?

PS: This occurred to me since we have the Leontief inverse (Understanding the Leontief inverse) as $(I-A)^{-1}=I+A+A^2+A^3+...$ so thought there might be something similar to this notion with the transpose.

statwoman
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    Matrix multiplication is associative, so this is $I + A + AA^T +A A^{T2} + AA^{T3} + ...$ – eyeballfrog May 24 '22 at 16:46
  • Using that so-called Leontif inverse the sum is $;I+A(I-A^T)^{-1};$ – greg May 24 '22 at 16:48
  • Let $B$ be that matrix $A'$ for an easy typing. Note that you can get rid of the many parentheses in the expression, e.g. $(AB)B$ is simply $ABB=AB^2$. Then you want to compute:$$I+A\underbrace{(I+B+B^2+B^3+\dots)}_{(I-B)^{-1}}\ ,$$ and the series of $B$-powers converges if $B$ is has (some) multiplicative norm $<1$, "same series" as in the question for $A$. – dan_fulea May 24 '22 at 20:18

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