I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove:
$||A - B|| \ge ||A^{-1}||^{-1}$
I dont understand why it makes a difference to say B is singular. Is there something special about the norm of a singular matrix? I know the condition number of B = inf and its determinate is 0, thus has no inverse, but I dont know how these things can help me determine the inequality above. I feel like I am missing a key point about singular matrices or something.
Also, would it be true to say in this instance that: $||A - B|| \ge ||A|| - ||B||$
or should that be less than or equal to instead of greater than or equal to.