# Eigenvectors from Eigenvalues

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Github repository

In “Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra” by Denton et al., the eigenvalue-eigenvector identity was (re)discovered. It can be rewritten in the following manner:

$\newline&space;\left&space;|&space;v_{i,j}&space;\right&space;|^{2}&space;=&space;\dfrac{\prod_{k=1}^{n-1}&space;(\lambda_{i}(H)&space;-&space;\lambda_{k}(h_{j}))}&space;{&space;\prod_{1&space;\leq&space;k&space;\leq&space;n}^{i&space;\neq&space;k}&space;(\lambda_{i}(H)&space;-&space;\lambda_{k}(H))}&space;\newline\newline\newline&space;{h_{j}}:&space;(n-1)\times(n-1)&space;\texttt{&space;matrix&space;with&space;jth&space;row&space;and&space;jth&space;column&space;removed.}&space;\newline&space;\lambda(H):&space;\texttt{&space;eigenvalues&space;of&space;}&space;H.&space;\newline&space;\lambda(h_{j}):&space;\texttt{&space;eigenvalues&space;of&space;}&space;h_{j}.$

This project reimplements the formula in both MATLAB and C++, throwing in some mischievous comparisons to MATLAB’s eig function, which also produces the eigenvectors of a given matrix. This was provoked by commentary from Cornell University’s Professor A. Townsend during a Linear Algebra lecture, which pondered the idea that this formula may indeed be useful in cases where only a few eigenvectors are needed from large matrices. While I cannot think of such a case in practice, it was an interesting idea to play with.