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Can we write explicitly the $(i,j)$ element of the matrix $$ (AA^T+I)^{-1} $$ where $A$ is an $n\times m$ matrix ?

In case that $i=j$ we can use the matrix inversion lemma to obtain the diagonal terms as $$ [(AA^T+I)^{-1}]_{1,1} = \frac{1}{1+b_1^H(B^HB+I)^{-1}b_1} $$ where we define $$ A =\left( \begin{array}{ccc} b_1^T \\ B \\ \end{array} \right) $$ Namely, $b_1^T$ is the first row of $A$.

I'm not sure if it is possible to obtain the off diagonal elements too.

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If everything else fails, Cramer's rule will give an explicit, if unwieldy, formula for each element as a quotient of polynomials in the entries of $A$. –  Henning Makholm Oct 21 '13 at 20:31

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