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st: "Diagonalizing" a non-symmetric matrix

From   Clyde Schechter <[email protected]>
To   [email protected]
Subject   st: "Diagonalizing" a non-symmetric matrix
Date   Thu, 02 Oct 2003 10:44:59 -0400

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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