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Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L+(ℝd, ℝd) be a continuous function taking values in the set ofd×d positive matrices. Denote by λM(x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula,$$\begin{gathered} \dim \left\{ {x \in \Sigma :\lambda _M \left( x \right) = \alpha } \right\} = \frac{1}{{\log m}} _{q \in \mathbb{R}}^{\inf } \left\{ { - \alpha q + P_M (q)} \right\} \hfill \\ = \frac{1}{{\log m}} _\mu ^{\max } \left\{ {h\left( \mu \right):M_ * \left( \mu \right) = \alpha } \right\} \hfill \\ \end{gathered} $$, where dim is the Hausdorff dimension or the packing dimension,PM(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and$$M_ * (\mu ) = \mathop {\lim }\limits_{x \to \infty } \frac{1}{n}\int {\log \left\| {M(y)M(\sigma y)...M(\sigma ^{n - 1} y)} \right\|d\mu \left( y \right)} $$.

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices