Let T be any square matrix and let denote any induced norm. Prove that
exists and equals
I am not sure how I go about proving that the limit exists.
For the infimum, I think it has something to do with the fact that
But I don't know how this information helps with the solution of the problem.
This is Gelfand's spectral radius formula (proof in the link).
[Thanks to Maddas for making the original post readable.]