I want to know the first order autocorrelation of a VAR(1) model. That can be calculated as follows $\rho(1)=\frac{\gamma(1)}{\gamma(0)}$ where $\gamma(1)$ denotes the covariance and $\gamma(0)$ denotes the variance.
A VAR(1) model can be described as follows: $$r_t=c+\Gamma r_{t-1}+\varepsilon_t$$ with $\varepsilon_t \sim N(0,\Sigma)$. Furthermore, $r_t$, $r_{t-1}$ and $\varepsilon_t$ all are a $N\times 1$ vector and the parameter matrix $\Gamma$ is a $N\times N$ matrix.
Taking the variance of $r_t$ gives us $\gamma(0) = Var(r_t) = Var(c+\Gamma r_{t-1}+\varepsilon_t)=\Gamma\times Var(r_{t-1})\times \Gamma' + \Sigma$.
So this gives us $\gamma(0) = \Gamma\times \gamma(0)\times \Gamma' + \Sigma$.
For $\gamma(1)$ I get $\gamma(1)=cov(r_t,r_{t-1})=cov(c+\Gamma r_{t-1}+\varepsilon_t,r_{t-1})=\Gamma cov(r_{t-1},r_{t-1})=\Gamma \gamma(0)$.
I just do not know how to implement the last part, i.e. $\rho(1)=\frac{\gamma(1)}{\gamma(0)}$, because of the matrices. I hope someone can help me with that?
I do know that the $\rho(1)$ of an AR(1) model is simply $\phi$ (i.e. the parameter of the lagged variable in an AR(1) model, see Finding the ACF of AR(1) process for instance). For its multivariate case, I also want to know the autocorrelation coefficient, hence this question.
I work with functional time series, in particular with the functional ARMA$(p,q)-$model. If you take a look e.g. at this publication http://arxiv.org/pdf/math/0509256v1.pdf from Mas (2005) or at "Function Spaces" a monographie from D. Bosq (2000) you can see how it works for the functional setting, hence for yours.
Division by $\gamma(0)$ doesn't work in your case, since it's a matrix. But you can consider divison by a matrix as multiplication of the inverse, if it does exist ($\text{det}(\gamma(0)) \neq 0 ?$).
I hope that's helpful.