Assume that $A \in R^{n\times m}$ is a matrix and I is the identity. I have a positive constant $\alpha$ which can be adjusted by me. I need to tune $\alpha$ in a way that $\Vert \alpha A^TA - I \Vert < 1$.
I know that if $A^TA$ is positive definite, it is doable. How about the time that $A^TA$ is positive semi-definite only?!
Any help would be appreciated.
No, this is not possible. Let $x_0 \in \mathrm{Ker}(A^T A) \setminus \{ 0 \}$, then $\frac{\| (\alpha A^T A - I)x_0 \|}{\| x_0 \|}=1$ for all $\alpha$ and an arbitrary vector norm, so if the matrix norm in question is induced by a vector norm then $\| \alpha A^T A - I \| \geq 1$.