Using Scipy's sparse.linalg.eigs package, one can find the $k$-largest eigenvalues and corresponding eigenvectors (each of size $n$) of an $n \times n$ matrix. I was wondering if it would be possible to use these eigenvalues/vectors to approximate the original matrix. For a full eigenvalue decomposition, one could do $A = V \Lambda V^{-1}$, where $V$ is the matrix of eigenvalues and $\Lambda$ is the diagonal matrix of eigenvalues. But since for the partial decomposition $V$ is of size $n \times k$, it is invertible. Still, is there any way to reconstruct $A$ from the largest eigenvalues/vectors?

(not sure if this helps, but in my case $A$ is Hermitian and relatively sparse)


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