Re: Eigenvalues/eigenvectors for a matrix of the form (A^T)A


<jcooper@xxxxxxxxxxx> wrote in message
news:1124398905.689060.309190@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>I have an m-dimensional real-valued symmetric matrix M which arises
> from the product (A^T)A where A is an nxm real-valued matrix which is
> not generally sparse. 'n' is much greater that 'm'. I would like to
> determine the eigenvalues and eigenvectors of M.
>
> The straightforward method would be to calculate M, then solve the
> eigenproblem. My problem is that M is strongly ill-conditioned, so
> that there are quantization problems in determining the eigenvalues.
> In the past, I have used the fact that the EVD (ie, eigenvalue
> decomposition) of 'M' is easily derived from the SVD of 'A' (but with
> the singular values of A being the square root of the eigenvalues of M
> -- goodbye quantization problems). Since n >> m, however, this results
> in an intermediate nxn matrix that I never really use anyway.
>
> It seems to me that this structure should lend itself to an efficient
> determination of the EVD of M without calculating M itself. Is there a
> good way to go about this without producing large, unused intermediate
> matrices?


One method is to do a QR decomposition by Householder or rotations: A = QR
where Q is orthogonal and R is an mxm upper triangular. Then EV's of R are
the same as EV's of A.


rusty.



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