Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma\_n$ the set of addmisible words with the length $n$. For any $n\in \mathbb{N},$ and $I \in \Sigma\_n$, we define
$$\|A(I)\|:=\sup\_{x\in [I]}\|A^{n}(x)\|.$$


We can also define the topological pressure for the potential $\log \|A\|$ as follows:


$$P(\log \|A\|):=\lim\_{n\to \infty} \frac{1}{n}\log \sum\_{|I|=n}\|A(I)\|.$$


Is it true that $P(\log \|A\|)$ can also be defined as follows:
$$P(\log \|A\|)=\lim\_{n\to \infty} \frac{1}{n}\log \sum\_{|I|\leq n}\|A(I)\|?$$



the definition of the topological pressure for matrices
Yes. This is a consequence of the following simple exercise about exponential growth: if $a\_n\geq 0$ is any sequence such that $P = \lim \frac 1n \log a\_n$ exists, then $\frac 1n \log \sum\_{k=1}^n a\_k \to P$ as well. (Apply this with $a\_n = \sum\_{|I|=n} \|A(I)\|$.)


To prove the exercise, observe that $\liminf \frac 1n \log \sum\_{k=0}^n a\_k \geq P$ is immediate, so it is enough to prove that $\limsup \frac 1n \log \sum\_{k=0}^n a\_k \leq P$. To do this, observe that for every $P' > P$ there is $C>0$ such that $a\_k \leq C e^{kP'}$ for all $k$, so $$\sum\_{k=0}^n a\_k \leq \sum\_{k=0}^n C e^{kP'} = C\frac{e^{(n+1)P'} - 1}{e^{P'} - 1}.$$


