what is the smallest upper bound for the following norm
$\|\left(\lambda\ I +A\ A^T\right)^{-1}\|<?$.
where, A is a rectangular matrix, $\lambda>0$ is a scalar. (any possible norm)
2026-03-30 05:10:12.1774847412
upper bound on a matrix norm
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We consider $||.||_2$. Let $\sigma^2$ be the smallest eigenvalue of $AA^T$ (this is linked to the singular values of $A$). Then $||(\lambda I+AA^T)^{-1}||_2=\dfrac{1}{\lambda+\sigma^2}$.
EDIT: Let $U=(\lambda I+AA^T)^{-1}$. Since $U$ is a symmetric $\geq 0$ matrix, $||U||_2$ is its greatest eigenvalue, that is $\dfrac{1}{\lambda+\sigma^2}$. Of course, $\sigma^2$ may be zero.