Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

Question

Consider some randomly generated matrix $B\in\mathbb{R}^{100\times100}$ and let $A:=BB^{\top}$

On MATLAB I computed the condition number of $A$, I obtained a value of $2.8377\mathrm{e}+04$

However if I apply the following modification to $A$ $$ A:=A+(\lambda_{min}(A)+10)I_{100}\tag{1} $$ Thus overall we have a symmetric positive-definite (SPD) matrix $A$.

When I run this on MATLAB, the condition number of $A$ is now reduced to $42.173$.

I know for once that originally when $A:=BB^{\top}$, the condion number of $A$ is the square of the condition number of $B$ and since $B$ is randomly generated, then its condition number should be ill and by squaring it, the condition number would further elevate. When applying this precondition technique in $(1)$, I have failed to establish an upper bound to the condition number of $A$ and thus:

Question: I can not seem to understand how this preconditioning technique had effectively reduced the condition number of $A$.

I would be grateful for any comments and/or answers!

Answer 1

If the eigenvalues of $A$ are $\lambda_1, \lambda_2, \dots,\lambda_n$, the eigenvalues of $A + \mu I$ are $\lambda_1 + \mu, \lambda_2 + \mu, \dots, \lambda_n + \mu$. It is an easy computation to verify that $$ \frac{\lambda_1 + \mu}{\lambda_n + \mu} < \frac{\lambda_1}{\lambda_n}, $$ when $\lambda_1 > \lambda_n>0$ and $\mu > 0$. In fact the LHS is a decreasing function of $\mu$.

Let me note, though, that this is not a preconditioning technique. It's just solving a different problem.

Answer 2

The accepted answer is right: you are not making a preconditioner.

To elaborate.

For a matrix $A$, a preconditioner is a matrix $B$ such that $B^{-1}A$ has a smaller condition number than $A$. The logic being that the preconditioned system $B^{-1}Ax=B^{-1}y$ is then easier to solve than $Ax=y$.

You're not doing this: you are finding a matrix $A'$ with a smaller condition number. That doesn't help you in solving a linear system with $A$.