I have a symmetric matrix $A$. Is there a way to have a positive semidefinite matrix $B$, so that solving a problem $X$ with $A$ or with $B$ is similar?
I thought about doing $\alpha \cdot I + A$, so that alpha is big enough. Does it work in all cases? ie find alpha such that $\alpha \cdot I + A$ is positive semidefinite?
Thank you in advance
If your matrix is real, then there is a basis in which it is diagonal. Just use $\alpha>\min_{\lambda\in\text{Sp}(A)}\lambda$ where $\text{Sp}$ denotes the spectrum of $A$ and you will have a semi definite positive matrix. However, this does not mean that the solution of your problem with this new matrix if "close" to the real solution. It depends on what is your problem at hand, and what you mean by "close" (a metric could be used to quantify this).