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From |
Kit Baum <[email protected]> |

To |
[email protected] |

Subject |
st: Re: eigenquery |

Date |
Fri, 3 Oct 2003 10:33:03 -0400 |

On Friday, Oct 3, 2003, at 02:33 US/Eastern, Pinaki wrote:

I would like to clarify my original problem. I have a couple of 88 by 88The diagonalization of a square (not necessarily symmetric) matrix involves both the eigenvalues and

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?

eigenvectors of the matrix. From de la Fuente, Mathematical methods and models for economists (Cambridge U Press, 2000):

Thm 6.7: Let A be an n x n matrix with n linearly independent eigenvectors. Then A is diagonalizable, with diagonalizing matrix E=[e_1, ..., e_n] whose columns are the eigenvectors of A and the resulting diagonal matrix is the matrix \Lambda = diag(\lambda_1, ... \lambda_n) with the eigenvalues of A in the principal diagonal, and zeros elsewhere. That is, E^{-1} A E = \Lambda.

Thm 6.8: Let A be an n x n matrix with n distinct eigenvalues. Then its eigenvectors are linearly independent, and A is diagonalizable.

The combination of these theorems suggests that if you run my program geneigen on a matrix A and get n (numerically) distinct eigenvalues, you may create the diagonal matrix referred to above with the eigenvalues on the diagonal, and call that the diagonalization of A. geneigen does not compute the eigenvectors because that is a hard problem, particularly so in a language that does not contain a complex data type. Of course, there is no guarantee that the n eigenvalues of the real general matrix A will have zero imaginary parts. If there are imaginary parts, then the theorem above implies that the diagonalizing matrix will not be real, but complex, and I don't know what you're going to do with that in Stata.

Kit

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