I saw in articles two different definitions for Lyapunov exponents of a discrete dynamical system.

Let's consider a discrete dynamical system $$ x_{k+1}=f(x_{k}),\quad x_{k}\in\mathbb{R}^{n},\quad k=0,1,2,\ldots, $$ $x_{0}$ -- given, where $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a smooth map.

Let $f'(x)$ be Jacobian of $f$ at $x$, $T$ denote the matrix transpose, $\lambda_{i}(\cdot)$ be the $i$th eigenvalue of a matrix. Let $\Phi_{m}=f'(x_{m-1})\ldots f'(x_{1})f'(x_{0})$.

Definition 1. The $i$th Lyapunov exponent at $x_{0}$ is given by $$ l_{i}(x_{0})=\lim_{m\to\infty}\frac{1}{2m}\ln|\lambda_{i}(\Phi_{m}^{T}\Phi_{m})|,\quad i=1,2,\ldots,n. $$

Definition 2. The $i$th Lyapunov exponent at $x_{0}$ is given by $$ l_{i}(x_{0})=\lim_{m\to\infty}\frac{1}{m}\ln|\lambda_{i}(\Phi_{m})|,\quad i=1,2,\ldots,n. $$

Are these definitions equivalent? If the answer is no, could you give a counterexample of $f$? And what is the correct definition of Lyapunov exponents?