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

From   "MITRA PINAKI (MAR1PXM)" <[email protected]>
To   [email protected]
Subject   st: RE: "Diagonalizing" a non-symmetric matrix
Date   Thu, 2 Oct 2003 12:15:35 -0400

I would like to clarify my original problem. I have a couple of 88 by 88
square matrices, some of them are symmetric and some are not. I would like
to diagonalize them. Now, the way I understand Matrix diagonalization is
that it is the process of taking a square matrix and converting it into a
diagonal matrix, where the entries in the diagonals are the eigen values. Is
this right?  

I would like to do this in Stata. Since I couldn't find any command in Stata
that can do diagonalization directly, I thought if I could find the eigen
values, I could construct a diagonalized matrix with the eigen values in the
diagonals and zeros in the off-diagonals. In Stata, symeigen and genigen
calculate eigen values for symmetric and non-symmetric matrices,
respectively. Is there anything wrong in this process? Is yes, could you
please recommend how can I do diagonalization in Stata?

Thank you very much for your help.


Pinaki Mitra

-----Original Message-----
From: Clyde Schechter [mailto:[email protected]] 
Sent: Thursday, October 02, 2003 10:45 AM
To: [email protected]
Subject: st: "Diagonalizing" a non-symmetric matrix

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.

Clyde Schechter

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