Unit Root
A autoregressive model AR(p) is usualy written as: $$ x_t = \sum_{i=1}^{p} \phi_i x_{t-i} + \epsilon_t $$
It can be rewritten using a matrix form where the state a time t is made of the last p observations:
$$ X_t = \Phi X_{t-1} + e_t $$ where
\begin{eqnarray} X_t = \begin{bmatrix} x_t \\ x_{t-1} \\ \vdots \\ x_{t-p} \end{bmatrix} , \Phi = \begin{bmatrix} \phi_1 & \phi_2 & \cdots & \phi_p \\ 1 & & & \\ & \ddots & & \\ & & 1 & 0 \\ \end{bmatrix} , e_t = \begin{bmatrix} \epsilon_t \\ 0 \\ \vdots \\ 0 \end{bmatrix} \end{eqnarray}
If all the eigenvalues in $\Phi$ matrix are smaller than one, X will decay with time rather than explode. The eigen values are the root of $z.Id-\Phi$ determinant. Testing for unit root mean testing that all roots of the characteristic polynomial are within unit circle. The polynom coefficients are well known for this expression from the Cayley-Hamilton Theorem:
\begin{eqnarray} \left| z.Id - \Phi \right| = \begin{vmatrix} z-\phi_1 & -\phi_2 & \cdots & -\phi_p \\ -1 & z & & \\ & \ddots & \ddots& \\ & & -1 & z \\ \end{vmatrix} = z^p - \sum_{i=1}^{p} \phi_i z^{p-i} \end{eqnarray}
Test for these roots are the ADF and Phillips-Perron tests - mean reversion can be interpreted as mean time of the pnl reversion - comparison variance at different time scale is expected to show a decreasing variance ratio