# How to recover Matrix ?

by Cylab
Tags: matrix, recover
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 No, you cannot "recover" the matrix from the eigenvalues alone for the simple reason that there exist an infinite number of matrics having the same eigenvalues. If D is a given diagonal matrix, and P is any invertible matrix, then $A= P^{-1}DP$ is a matrix having the numbers on D's diagonal as eigenvalues. Different P matrices will, in general, give different matrices having the same eigenvalues.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 Again, no. for example, the matrices $$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$ $$\begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}$$ $$\begin{bmatrix}6 & 0 \\ 0 & -1\end{bmatrix}$$ and, generally, $$\begin{bmatrix}X & 0 \\ 0 & Y\end{bmatrix}$$ for any x and y, all have i and j as eigenvectors but are different matrices with, of course, different eigenvalues. To point out what should be obvious, two different matrices can have exactly the same eigenvalues and corresponding eigenvectors. In that case, they would be similar matrices.