Square matrix and induced norm

**math8** · April 10th 2010, 10:44 AM

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.

**maddas** · April 10th 2010, 12:54 PM

This isn't an answer; I'm just retyping your equations which are broken for me (hotlinking I think)...

Originally Posted by math8

Let T be any square matrix and let $||\cdot||$ denote any induced norm. Prove that $\lim_{n\to\infty}||T^n||^{1/n}$ exists and equals $\inf_{n=1,2,\cdots} ||T^n||^{1/n}$ .

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

$||T^n|| = \sup_{x\neq0} {||T^nx||\over||x||}$

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

**Opalg** · April 11th 2010, 12:22 AM

Originally Posted by maddas

This isn't an answer; I'm just retyping your equations which are broken for me (hotlinking I think)...

Originally Posted by math8
Let T be any square matrix and let $\|\cdot\|$ denote any induced norm. Prove that $\lim_{n\to\infty}\|T^n\|^{1/n}$ exists and equals $\inf_{n=1,2,\ldots} \|T^n\|^{1/n}.$

This is Gelfand's spectral radius formula (proof in the link).

[Thanks to Maddas for making the original post readable.]

**math8** · April 12th 2010, 01:26 PM

Thanks to Maddas and to Opalg.

I can understand the proof of the existence of $lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ .

$lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ exists and is equal to $\rho (T)$ , the spectral radius of T.

But how do we show that $\rho (T) = inf_{n=1,2, \cdots } \left\| T^{n} \right\| ^{1/n}$ ?

**Opalg** · April 13th 2010, 12:17 AM

Originally Posted by math8

Thanks to Maddas and to Opalg.

I can understand the proof of the existence of $lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ .

$lim_{n \rightarrow \infty }\left\| T^{n} \right\| ^{1/n}$ exists and is equal to $\rho (T)$ , the spectral radius of T.

But how do we show that $\rho (T) = inf_{n=1,2, \cdots } \left\| T^{n} \right\| ^{1/n}$ ?

This follows from the fact the norm has to be submultiplicative, so that $\|S^2\|\leqslant\|S\|^2$ for any matrix S.

Suppose that, for some n, $\|T^n\|^{1/n}=\lambda<\rho(T)$ . Then $\|T^n\|=\lambda^n$ , $\|T^{2n}\| = \|(T^n)^2\| \leqslant \|T^n\|^2= \lambda^{2n}$ , and by induction $\|T^{2^kn}\|\leqslant\lambda^{2^kn}$ for all k.

Therefore $\|T^{2^kn}\|^{1/(2^kn)}\leqslant\lambda$ . let $k\to\infty$ to see that $\rho(t)\leqslant\lambda$ — contradiction.

**math8** · April 14th 2010, 06:35 AM

I might be wrong but according to that proof, does this just mean that $\rho (T)$ must be a lower bound for $\{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ ?

How do we show that $\rho (T)$ is actually the greatest lower bound? I mean is there an n such that $\rho (T) = \left\| T^{n} \right\| ^{1/n}$ ?

**Opalg** · April 14th 2010, 08:43 AM

Originally Posted by math8

I might be wrong but according to that proof, does this just mean that $\rho (T)$ must be a lower bound for $\{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ ?

How do we show that $\rho (T)$ is actually the greatest lower bound? I mean is there an n such that $\rho (T) = \left\| T^{n} \right\| ^{1/n}$ ?

Yes, it shows that $\rho (T)$ must be a lower bound for $\{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ . But I thought you had already shown that $\rho (T) = \lim_{n\to\infty} \left\| T^{n} \right\| ^{1/n}$ . Taken together, those results show that $\rho (T)$ is the greatest lower bound.

**math8** · April 14th 2010, 01:56 PM

Oh, I didn't realize that. I actually thought that for that result, one actually has to show that there is an n such that $\rho (T) = \left\| T^{n} \right\| ^{1/n}$ . (without the limit)

But I guess you're right.

Thanks a lot!

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