Can there be two matrices that have the same eigenvalues and corresponding eigenvectors? Lets say there is a matrix with eigenvalues 1 and 2 and eigenvectors $(1,2)^T$ and $(3,4)^T$, will there be another matrix with these? I want to intuitively say no, but I'm not positive.
[Math] Is a matrix with certain eigenvectors and eigenvalues unique
eigenvalues-eigenvectorslinear algebra
Best Answer
It is possible for multiple matrices to have the same eigenvalues and eigenvectors as long as the diagonals have the same entries, but in a different order.
If the elements of the trace are equal, and the traces are equivalent, then the eigenvalues and eigenvectors will be the same.