Can a Matrix be Recovered from Eigenvalues Alone?

[Cylab]

Hello

Is it possible to recover Matrix from eigenvalue alone?
that is, A = PDP^-1,,,
once only D (eigenvalues) is known,, without knowing eigenvectors,
is it possible to recover A?

Thanks

P.S. I will appreciate if you can provide me with some algorithms about recovering original matrix..:)

 

[HallsofIvy]

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.

 

[AlephZero]

It should be obvious that you can't do this just be counting the number of terms involved.

An n x n matrix has n² terms which are all indepdent of each other. You can't re-create n² different numbers from just n eigenvalues, unless n = 1.

 

[Cylab]

Thanks a lot for your attention. They are really helpful.
How about opposite, that is,
Is it possible to recover Matrix from eigenspace alone (without knowing eigenvalues)?
Or does each erigenvector reveal some information?

Thanks again.

 

[HallsofIvy]

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.