# Is a matrix with certain eigenvectors and eigenvalues unique?

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.

• The two $2 \times 2$ diagonal matrices $diag(1,2)$ and $diag(2,1)$ have the same eigenvalues and eigenvectors, just in a different order. – Simon S Apr 9 '15 at 14:26

• I tried this with my example, by solving $BDB^{-1}$ where B has the eigenvectors as columns and D with the eigenvalues as corresponding diagonal values. I then did the same with the eigenvalues and eigenvectors switched. Both gave me the same matrix, which I believe would make it unique. Or is this just a special case? – TheStrangeQuark Apr 9 '15 at 14:43