Re: minimization of a matrix function

On Jul 2, 11:58 am, spellu...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
(Peter Spellucci) wrote:
In article <1183331109.000116.173...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, hs.sa...@xxxxxxxxx writes:

>On Jun 30, 12:54 pm, Ron Shepard <ron-shep...@xxxxxxxxxxxxxxxxxx>
>wrote:
>> In article <1183146043.639165.208...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>>
>>
>>
>> hs.sa...@xxxxxxxxx wrote:
>> > Hello,
>>
>> > I have a situation where I need to solve for vectors x_i, where X =
>> > [x_1 x_2 . . x_d] such that
>>
>> > XQX' = diag{a_1, a_2, ..., a_d}
you confused X and X' here: in your notation X is N times d
hence you must build X'QX.

Yes, thanks for the correction.


you also forgot to mention that you require X'X = I_d,
which gives pretty disagreeable side constraints

hm .. yes, that's true too.


>>
>> > where Q is a square, real and symetric NxNmatrixand X' denotes
>> > transpose of X and a_i's are scalars. Also, d << N.
>>
>> > I see that one way is tominimizethe following:
>>
>> > XQX' - diag{a_1, a_2, ..., a_d}
>>
>> > to obtain the solution X*.
>>
<SNIP>
as long as the eigenvalues stay disjoint, you could
solve the eigenvalue problem first for a given system and then
use Newtons method for the nonlinear system

[ Q x_i ] [ x_i ] = [ 0 ]
[ x_i' 0 ] [ a_i]= [ 1 ]

Shouldn't that be:

[ Q -x_i ] [ x_i ] = [ 0 ]
[ x_i' 0 ] [ a_i ] = [ 1 ]

where x_i is a Nx1 vector, Q is NxN matrix, and a_i is a scalar? This
is because the top equation should be such that:
Q x_i = a_i x_i

and the bottom equation should be
x_i' x_ = 1.


which would converge fast if Q has changed only little, taking the "old"
x_i, a_i as initial guess. orthogonality of eigenvectors for different
eigenvalues of the symmetric Q is automatic

So you mean I can solve for each x_i separately using an optimization
algorithm? In other words, I don't have to specify the constraint x_i'
x_j = 0 (where i and j are distinct)?

thanks for you comments and help,
regards,
->HS



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