I have a formula for the value of $\phi$ such the a matrix $A = A(\phi)$ becomes singular.
To check this formula I run a simulation in Matlab where I provide $A$ with a range of values of $\phi$ and observe for which $\phi$ the minimum eigenvalue of $A$ is closest to zero.
The thing is, if I just use $A$ 'as is' in the simulation code I don't get a good match between the formula and the simulation. However if I use $A^\ast A$ instead (which is singular when $A$ is singular) I get a very good match.
I believe I recall once hearing something about $A^\ast A$ being more numerically stable than $A$, and that $A^\ast A$ is a 'better' matrix than $A$.
So why am I getting much better results using $A^\ast A$? Can anyone shed some light on the potential reasons for this?