Let $Z_n^x =(Z^x_n (1), \cdots, Z^x_n (d))$ be a supercritical d-type branching process in an independent and identically distributed random environment $(\xi_n)$, starting with $Z_0=x \in \bb N^d\setminus \{0\}$, whose offspring distribution at time $n$ depends the environment $\xi_n$. Let $M_n = M(\xi_n) $ be the mean matrix of the offspring distribution at time $n$. We establish a Kesten-Stigum type theorem for the scalar product $\langle Z_n^x, y \rangle $ for any $y \in \bb R_+^d\setminus \{0\}$: under suitable conditions, $W_n^x (y):= \langle Z_n^x, y \rangle / \langle x M_0 \cdots M_{n-1}, y \rangle $ converges in probability to some $\bb R_+$-valued random variable $W^x$; the almost sure convergence is also established under additional moment conditions; a criterion is given for $W^x$ to be non-degenerate. In the proof, we find $(u_n)$ such that $(W_n^x(u_n))$ is a martingale, and prove that $W_n^x (y)$ converges uniformly for $y \in \bb R_+^d\setminus \{0\}$ to the limit $W^x$ of $W_n^x(u_n)$. We also prove a duality of the Kesten-Stigum theorem about the convergence of $ \langle Z_{n+k}, y \rangle/ \langle Z_n M_n \cdots M_{n+k-1}, y \rangle$ for fixed $k$ as $n\to \infty$, and a theorem about the convergence of the direction $Z_n^x/ \|Z_n^x\|$. An important ingredient of the proofs is the Perron-Frobenius type theorem that we establish for the products of random matrices, which is of independent interest. Let $\{M_n: n\in \bb Z \} $ be a stationary and ergodic sequence of positive random matrices, and let $M_{k,n} = M_k \cdots M_n$, for $k\leq n$. We find unit vectors $u_n,v_n >0$, and scalars $\lambda_n, \mu_n, a_{k,n} >0$, such that, almost surely, for fixed $k$ as $n\rightarrow \infty$, and for fixed $n$ as $k\rightarrow -\infty$, $ \langle x M_0 \cdots M_{n}, y \rangle \sim a_{k,n} \langle u_k, x \rangle \langle v_{n}, y \rangle $ uniformly in $x$ and $y$, where $a_{k,n}$ can be taken as the product form $a_k \mu_k \cdots \mu_n$; all the sequences $(u_k)$, $(v_n)$, $(\lambda_n)$, $(\mu_n)$ and $(a_k)$ are stationary and ergodic. As further applications, we find new laws of large numbers and central limit theorems for $\langle Z_n^x, y\rangle$, as well as for the products of random matrices.