Let $(X,\mathcal B,\mu)$ be a probability space, let $T:X\to X$ be an invertible ergodic measure-preserving map, and let $A:X\to GL(d,\mathbb R)$ be measurable with \begin{align*} \int_X \log^+\|A(x)\|\,d\mu(x)<\infty, \end{align*} and \begin{align*} \int_X \log^+\|A(x)^{-1}\|\,d\mu(x)<\infty. \end{align*} Then there exist [real numbers](/page/Real%20Numbers) \begin{align*} \lambda_1>\lambda_2>\cdots>\lambda_k \end{align*} and, for $\mu$-a.e. $x\in X$, a direct-sum decomposition \begin{align*} \mathbb R^d=E_1(x)\oplus\cdots\oplus E_k(x) \end{align*} such that $A(x)E_i(x)=E_i(Tx)$ and for every $v\in E_i(x)\setminus\{0\}$, \begin{align*} \lim_{n\to\pm\infty}\frac{1}{n}\log |A^n(x)v|=\lambda_i. \end{align*} For $n<0$, the convention is \begin{align*} A^n(x)=A^{-n}(T^n x)^{-1}, \end{align*} so the negative iterates use the inverse cocycle along the backward orbit of the invertible base map, and the dimensions $m_i=\dim E_i(x)$ are constant for $\mu$-a.e. $x$.