# proof of Gelfand spectral radius theorem

For any $\epsilon>0$, consider the matrix

 $\tilde{A}=(\rho(A)+\epsilon)^{-1}A$

Then, obviously,

 $\rho(\tilde{A})=\frac{\rho(A)}{\rho(A)+\epsilon}<1$

and, by a well-known result on convergence of matrix powers,

 $\lim_{k\to\infty}\tilde{A}^{k}=0.$

That means, by sequence limit definition, a natural number $N_{1}\in\mathbf{N}$ exists such that

 $\forall k\geq N_{1}\Rightarrow\|\tilde{A}^{k}\|<1$

which in turn means:

 $\forall k\geq N_{1}\Rightarrow\|A^{k}\|<(\rho(A)+\epsilon)^{k}$

or

 $\forall k\geq N_{1}\Rightarrow\|A^{k}\|^{1/k}<(\rho(A)+\epsilon).$

Let’s now consider the matrix

 $\check{A}=(\rho(A)-\epsilon)^{-1}A$

Then, obviously,

 $\rho(\check{A})=\frac{\rho(A)}{\rho(A)-\epsilon}>1$

and so, by the same convergence theorem,$\|\check{A}^{k}\|$ is not bounded. This means a natural number $N_{2}\in\mathbf{N}$ exists such that

 $\forall k\geq N_{2}\Rightarrow\|\check{A}^{k}\|>1$

which in turn means:

 $\forall k\geq N_{2}\Rightarrow\|A^{k}\|>(\rho(A)-\epsilon)^{k}$

or

 $\forall k\geq N_{2}\Rightarrow\|A^{k}\|^{1/k}>(\rho(A)-\epsilon).$

Taking $N:=max(N_{1},N_{2})$ and putting it all together, we obtain:

 $\forall\epsilon>0,\exists N\in\mathbb{N}:\forall k\geq N\Rightarrow\rho(A)-% \epsilon<\|A^{k}\|^{1/k}<\rho(A)+\epsilon$

which, by definition, is

 $\lim_{k\to\infty}\|A^{k}\|^{1/k}=\rho(A).\,\,\square$

Actually, in case the norm is self-consistent (http://planetmath.org/SelfConsistentMatrixNorm), the proof shows more than the thesis; in fact, using the fact that $|\lambda|\leq\rho(A)$, we can replace in the limit definition the left lower bound with the spectral radius itself and write more precisely:

 $\forall\epsilon>0,\exists N\in\mathbb{N}:\forall k\geq N\Rightarrow\rho(A)\leq% \|A^{k}\|^{1/k}<\rho(A)+\epsilon$

which, by definition, is

 $\lim_{k\to\infty}\|A^{k}\|^{1/k}=\rho(A)^{+}.$
Title proof of Gelfand spectral radius theorem ProofOfGelfandSpectralRadiusTheorem 2013-03-22 15:33:55 2013-03-22 15:33:55 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 7 Andrea Ambrosio (7332) Proof msc 34L05