# Homework Help: Induced norms

1. Apr 10, 2010

### math8

Let T be any square matrix and let $$\left\| \cdot \right\|$$ denote any induced norm. Prove that

$$lim_{n \rightarrow _{\infty}} \left\| T^{n} \right\| ^{1/n}$$ exists and equals $$inf _{n=1,2,\cdots } \left\| T^{n} \right\| ^{1/n}$$

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

$$\left\| T^{n} \right\| = sup_{x \neq 0} \frac{\left\| T^{n} x\right\|}{\left\| x \right\|}$$

But I don't know how this information helps with the solution of the problem.