What are eigenvectors and eigenvalues in layman terms? I am learning Singular Vector Decomposition (SVD) technique. It breaks a matrix X into 3 matrices U, S and $V^T$. U is formed by eigenvectors of matrix X. 
My understanding is that eigenvectors are the independent components of a matrix using which other components can be formed. Am I correct? 
Can we recreate the parent matrix just by knowing its eigenvectors?
 A: Can we recreate the parent matrix $\mathbf{A}$ just by knowing its eigenvectors? No.
Can we recreate the parent matrix $\mathbf{A}$ just by the eigenvectors of $\mathbf{A}^{*}\mathbf{A}$? No.

Start with a matrix
$$
\mathbf{A} \in \mathbb{C}^{m\times n}_{\rho}.
$$
The Fundamental Theorem of Linear Algebra can be represented by the orthogonal decomposition of the domain $\mathbb{C}^{n}$ and the codomain $\mathbb{C}^{m}$:
$$
\begin{align}
%
  \mathbb{C}^{n} &= 
    \color{blue}{\mathcal{R}\left(\mathbf{A}^{*}\right)} \otimes
    \color{red} {\mathcal{N}\left(\mathbf{A}\right)} \\
%
 \mathbb{C}^{m} &= 
    \color{blue}{\mathcal{R}\left(\mathbf{A}\right)} \otimes
    \color{red} {\mathcal{N}\left(\mathbf{A}^{*}\right)} \\
%
\end{align}
$$

THe beauty of the singular value decomposition 
$$
  \mathbf{A} = \mathbf{U} \, \Sigma \, \mathbf{V}^{*}
$$
is that it resolves the four fundamental subspaces.
To construct the SVD, start by computing the eigenvalue spectrum of the product matrix:
$$
 \lambda \left( \mathbf{A}^{*} \mathbf{A} \right). 
$$
The eigenvales are nonnegative. Order them, and remove the $0$ values creating the set $\hat{\lambda}$. The singular values are
$$
  \sigma = \sqrt{\hat{\lambda}}.
$$
The normalized eigenvectors of the product matrix are the first $\rho$ columns of the matrix $\mathbf{V}$ and are a span of the range space $\color{blue}{\mathcal{R}\left(\mathbf{A}^{*}\right)}$. The remaining $n-\rho$ columns are an orthonormal span for the null space $\color{red}{\mathcal{N}\left( \mathbf{A} \right)}$.
The domain matrix is
$$
\mathbf{V} = 
\left[ \begin{array}{cc}
  \color{blue}{\mathbf{V}_{\mathcal{R}}} &
  \color{red} {\mathbf{V}_{\mathcal{N}}}
\end{array} \right]
%
=
\left[ \begin{array}{ccc|ccc}
  \color{blue}{v_{1}} & \dots & \color{blue}{v_{\rho}} &
  \color{red}{v_{\rho+1}} & \dots & \color{red}{v_{n}}
\end{array} \right]
%
$$
Again, these vector space the domain:
$$
  \color{blue}{\mathcal{R}\left(\mathbf{A}^{*} \right)} = 
\text{span } 
\left\{
\color{blue}{u_{1}}, \dots, \color{blue}{u_\rho}
\right\}, \qquad
%
\color{red}{\mathcal{R}\left(\mathbf{A} \right)} = 
\text{span } 
\left\{
\color{red}{u_{\rho+1}}, \dots, \color{red}{u_{n}}
\right\}
$$
The domain is well characterized, yet the codomain is completely unknown. We don't have enough information to construct $\mathbf{A}$ if we only have $\mathbf{V}$ and $\Sigma$.
