I am trying to reconstruct a symmetric 3 x 3 matrix from just its eigenvalues and eigenvectors.
I think the solution involves orthogonalizing two of the eigenvectors using the Gram-Schmidt procedure, but I am not sure why or how. Thanks.
[Math] How to reconstruct a symmetric matrix given the eigenvalues and eigenvectors
eigenvalues-eigenvectorslinear algebramatricessymmetric matrices
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Best Answer
If the columns of a matrix $P$ are three linearly independent eigenvectors of a matrix $A$ corresponding to the eigenvalues $a,b,c$ respectively, then $P\operatorname{diag}(a,b,c)P^{-1}$ must be equal to $A$, regardless of whether $P$ is orthogonal or not. You don't need any orthogonalisation process. The resulting matrix is automatically symmetric (except that there are some rounding errors, of course, if you use floating point compuatation instead of symbolic computation) when $A$ is symmetric.