How to recover Matrix ?

  1. 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?


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

    HallsofIvy 40,372
    Staff Emeritus
    Science Advisor

    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 [itex]A= P^{-1}DP[/itex] 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.
  4. AlephZero

    AlephZero 7,300
    Science Advisor
    Homework Helper

    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.
  5. 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.
  6. HallsofIvy

    HallsofIvy 40,372
    Staff Emeritus
    Science Advisor

    Again, no. for example, the matrices
    [tex]\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}[/tex]
    [tex]\begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}[/tex]
    [tex]\begin{bmatrix}6 & 0 \\ 0 & -1\end{bmatrix}[/tex]
    and, generally,
    [tex]\begin{bmatrix}X & 0 \\ 0 & Y\end{bmatrix}[/tex]
    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.
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