Let's say we have a full rank square matrix $A$ of size $n$ and we have found the eigenvectors of the matrix through eigen-decomposition/SVD.
Here, $A = [ V_1 , V_2 , V_3 , \ldots, V_n ]$ , where $V_i$ is a vector of size $n\times 1$.
How can I express $V_i$ as a linear combination of eigenvectors of $A$?
If $A$ ist normal then by the spectral theorem $A=UDU^T$ where $D$ is a diagonal matrix containing the eigenvalues and $U$ is orthonormal matrix whose columns are the eigenvectors.
Then all you have to do is solve the linear system $UX=A$ since then then the first column if $A$ equals a linear combination of the columns of $U$ whose coefficients are given by the first column of $X$, etc.
Since $U$ is orthonormal, the system is trivial to solve: $X= U^T A=DU^T$. So the Eigendecomposition already encodes this information.