For a given matrix $H$, if there is a vector $V$ $$ H V = \lambda V $$,
where $\lambda$ is a constant, then the vector $V$ is called the eigen vector of the matrix $H$, and the $\lambda$ is the eigenvalue.
Suppose an arbitray vector $V$ defined within a space of dimension $n$, formed by a set of orthogonormal basis vectors $\{V_1, V_2, \cdots, V_n\}$, which are the eigenvectors of the matrix $H$,
$$ HV_n = \lambda_n V_n, \quad \lambda_1 > \lambda_2 > \cdots > \lambda_n, $$
then
$$ V = \sum_n c_n V_n, \quad V^T_n V_{n'} = \delta_{nn'} $$
If one keep multiplying the arbitrary vector $V$ with the matrix $H$ many times,
$$ H^m V = H^m \sum_nc_n V_n = \sum_nc_n \lambda^m_n V_n. $$
The above expression indicates that each component (eiven vectors) $V_n$ in the $V$ is scaled with the eigenvalue $\lambda^m_n$. For simplicity, we define $U^{(m)} = H^m V$. If one normalized the $U^{(m)}$,
$$ \widetilde U^{(m)} = \dfrac{U^{(m)}}{U^{(m)T} U^{(m)}} = \dfrac{\sum_nc_n \lambda^m_n V_n}{\sum_{nn'}c^\ast_nc_{n'} \lambda^m_{n'} V^T_{n'} \lambda^m_n V_n} =\dfrac{\sum_nc_n \lambda^m_n V_n}{\sum_{nn'} |c_n|^2 \lambda^{2m}_{n} } =\dfrac{\sum_nc_n (\lambda_n/\lambda_1)^m }{\sum_{nn'} |c_n|^2 (\lambda_n/\lambda_1)^m }V_n. $$
Since $\lambda_n/\lambda_1=1$ for $n=1$ and $<1$ for $n\ne 1$, as long as $m$ is sufficient large, one has
$$ \widetilde U^{(m)} \sim \dfrac{c_1 }{ |c_1|^2}V_1. $$
In physics, many methods are baed on the above idea to get the ground state of systems, including imaginary time evolution.
Starting from a trivial wave function $\vert\Psi\rangle$ which is not orthongonal to the groud state, then one can always expand this wave function in terms of the eigenvectors of the Hamiltonian $H$, $$ \vert\Psi\rangle = \sum_n c_n \vert \Phi_n\rangle,\quad H \vert \Phi_n\rangle = E_n \vert \Phi_n\rangle, $$
Applying the following operator onto the initial wave function can bring us to the ground state $$ e^{-H\tau} \vert \Psi\rangle = \sum_n \dfrac{(-H\tau)^n}{n!}\vert \Psi\rangle \to \vert \Phi_1\rangle. $$
An intuitive picture can be found here: http://setosa.io/ev/eigenvectors-and-eigenvalues/