# st: RE: "Diagonalizing" a non-symmetric matrix

 From "Nick Cox" <[email protected]> To <[email protected]> Subject st: RE: "Diagonalizing" a non-symmetric matrix Date Thu, 2 Oct 2003 16:06:37 +0100

```I understood Mitra to want a matrix with
the eigenvalues along the diagonal. That's
what was asked for. It's true, as I understand
it, that diagonalization as such is a different
thing.

Of course, if the eigenvalues have imaginary
parts it is rather difficult to do anything
with that matrix in Stata, but I didn't think it
necessary to spell that out. Either way, -geneigen-
would seem to be the first step in Stata, even
if one can't go beyond it.

Nick
[email protected]

> -----Original Message-----
> Clyde Schechter
>
> Yesterday, Mitra Pinaki asked for help in diagonalizing a
> non-symmetric
> matrix.  Nick Cox pointed him to Kit Baum's -geneigen-
> which will, indeed,
> calculate the eigenvalues of a non-symmetric matrix.  But that won't
> diagonalize the matrix.  In fact, it's fairly simple linear
> algebra to show
> that any matrix of real numbers which _can_ be diagonalized
> is necessarily
> symmetric!  So it's unclear what M.P. is trying to do.
>
> The most that can be done with a non-symmetric matrix is to
> identify its
> eigenvalues and eigenvectors (which, due to non-symmetry, will have
> cardinality strictly less than the rank of the matrix) and
> then create a
> diagonal matrix out of those.  But that diagonal matrix
> will be of smaller
> rank than the original, and is not a conjugate of the
> original matrix.  Any
> conjugate of the original which does diagonalize as much as
> possible will
> still have a piece which has non-zero elements off the diagonal.

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```