Eigenvalues, stupid

[Kamil Toth <kamiltoth@yahoo.com>]

Dear S-PLUS gurus:

I have a linear algebra question, may be trivial, may be not. Indirectly, it is related to the accuracy of S-PLUS computation of eigenvalues, using the function "eigen".

 

Suppose that I have a non-symmetric, positive definite matrix A, and I compute its eigenvalues, {lmb1,...,lmbN}, using eigen(A).  Suppose now that I have a new matrix

C=diag{b1,...,bN}%*%A. Just to make things more clear, I multiply the matrix A by a vector {b1,...,bN} in such a way that the first row of A is multiplied by b1, the second by b2, ....., etc. All b's are positive.

 

Question: Is that possible to say anything about the eigenvalues of matrix C not actually computing them as eigen(C)?

 

The reason I am asking is this: in my case,  A is a reasonably behaving matrix, with spectrum {lmb} in unit circle. But vector B is a terrible thing with values in the range of many orders of magnitude. In addition, typical orders N are in hundreds.

A couple of artificially constructed examples show that determination of eigenvalues

of C using eigen(C) may be terribly wrong. In this situation, I would rather prefer to compute the eigenvalues of C through those of A, if linear algebra has anything to say about such a problem.

 

Thank you in advance

 

Kamil Toth

 

 

 

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