Eigenvectors from Eigenvalues

Published: January 01, 2019

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:

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

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.

Share on

Twitter Facebook LinkedIn