Re: generating random matrix with negative eigenvalues
- From: ellieandrogerxyzzy@xxxxxxxxxxxxxxxxxxxxxx (Roger Stafford)
- Date: Wed, 02 Aug 2006 21:50:30 GMT
In article <ef3ccda.6@xxxxxxxxxxxxxxxxxxxxxxx>, "Cenny Taslim"
<usro@xxxxxxxxxxxx> wrote:
....------------------------
I just tried your method and it seemed to be what I wanted. Although
like you pointed out, I wasn't sure how random the resulting matrices
are. Would I occasionally get symmetric/asymmetric matrices. What
about matrices with negative eigenvalues that have imaginary parts
also? Should I produce them the same way?
....
Cenny
With D a diagonal matrix and V a real unitary matrix (as generated by
orth(randn(n)) ), then A = V*D*V' will always yield a symmetric matrix,
and its eigenvalues will be the same as those of D, namely its diagonal
elements. This remains true even if some of the elements of D are
complex-valued. You can easily demonstrate this by taking the
non-conjugate transpose of both sides of the equation A = V*D*V' and
noting that the right side remains unchanged if V is real and D is a
diagonal matrix.
The worry I alluded to with respect to 'orth' pertains essentially to
concerns about how much dependence there is in orth's algorithm on the
ordering that is present in the columns or rows of its argument matrix.
It may conceivably act preferentially on the earlier columns or rows as
the algorithm proceeds. If you begin with a random ordering of the
eigenvalues in D (say, by using 'randperm'), I am reasonably sure that you
will obtain a random (uniform) sampling of A matrices possessing that
given set of eigenvalues. On the other hand, if, say, the D values were
to be sorted by size, then I am not so sure about the randomness of the
resulting A = V*D*V' for the given set.
Roger Stafford
.
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- generating random matrix with negative eigenvalues
- From: Cenny Taslim
- Re: generating random matrix with negative eigenvalues
- From: Cenny Taslim
- generating random matrix with negative eigenvalues
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