Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Re: Minimizing matrix norm
Posted:
Jul 3, 2006 8:19 AM
|
|
On 2006-07-03 05:07:45 -0300, Ronald Bruck <bruck@math.usc.edu> said:
> I have the following problem: given n x n symmetric real matrices
>
> A, F1, ..., Fm,
>
> I want to minimize the matrix 2-norm of
>
> ||A - \sum_i c_i F_i||, c_i \in R
For some purposes it might make sense to use the Frobenious norm rather
than the L_2 norm. Much simpler expression. It is a matrix norm but just
not induced by, or compatible with in other terminology, a vector norm.
One description is that it treats the matrix as if it were a vector rather
than an operator.
> (i.e. the norm of A as an operator from R^n to R^n with the usual
> Euclidean norms).
>
> I know how to use SDP to minimize the maximum eigenvalue of
>
> A - \sum_i c_i F_i.
>
> What I've been doing is doubling the dimension, replacing A by the block
> matrix
>
> ( A 0 )
> ( 0 -A )
>
> and the Fi by
>
> ( Fi 0 )
> ( 0 -Fi)
>
> and minimizing the maximum eigenvalue of the new problem. (It isn't
> quite as bad as it sounds, because SDP solvers can usually take
> advantage of block structure like this.)
>
> This works, but am I missing something? Is there a more direct way?
|
|
|
|
