Can Eigenvalues and Determinants Determine the Elements of a Square Matrix?

[Cygni]

Hello,

Given that we have some matrix M with unknown real elements a, b, c, d and we know its eigenvalues \lambda_{1} and \lambda_{1} (no eigenvectors though) and the value of the determinant is it possible to find the elements and hence the matrix M using this informaiton?

Thanks

 

[Petr Mugver]

No, there are infinite matrices with the same eigenvalues, but different eigenvectors. For example

\left(\begin{array}{cc}1&2\\0&3\end{array}\right)\qquad\textrm{and}\qquad\left(\begin{array}{cc}3&4\\0&1\end{array}\right)

have the same eigenvalues, but are...different!

By the way, the determinant is just the product of the eigenvalues, so it doesn't give further information.

 

[Simon_Tyler]

I think that the below is basically correct...

Provided you have n eigenvalues for the nxn matrix, then if you know the eigenvalues then you know the matrix up to a similarity transformation.

If you also know the eigenvectors then that gives you the similarity transformation and thus you know the matrix up to permutation of rows. (provided the eigenvectors are distinct)

 

[Petr Mugver]

If you also know the eigenvectors then that gives you the similarity transformation and thus you know the matrix up to permutation of rows. (provided the eigenvectors are distinct)

Actually in this case you know the matrix exactly, without any possibility of interchanging rows or columns, because if you change the order of the eigenvalues you have to change the order of the eigenvectors as well, and the result is always the same matrix.

 

[Simon_Tyler]

@Petr

I knew (and have taught) that!
That should teach me not to post whilst watching tv.