Low-Dimensional Matrix Approximation

[jfy4]

Hi,

Lets say that I have a 4x4 matrix, and am interested in projecting out the most important information in that matrix into a 2x2 matrix. Is there an optimal projection to a lower dimensional matrix where one keeps most of the matrix intact as best as possible? Thanks.

 

[Number Nine]

What information are you talking about, exactly? 4x4 and 2x2 matrices are rather different (they're functions on completely different spaces). If there's a particular (say, 2-dimensional) subspace of interest, then the restriction of a 4x4 matrix to the subspace may give you a 2x2 matrix, but we really can't say anything specific without more information.

 

[micromass]

I think you should be more specifc. What do you want it for? Do you have an example? Etc.

I can easily think of a (nonlinear) way to transform a $4\times 4$ into a $2\times 2$ such that all information is preserved, but I doubt you're looking for this.

 

[jfy4]

good questions. True be told I'm not entirely sure... The matrices house local configurations on a lattice, and I can't keep them all due to computer memory cost, so I want to truncate the matrices and keep as much as I can. The only way I have been doing it is with SVD. Once I preform the SVD I rotate the matrix using the V ( from U λ V$^{\dagger}$) and then I project out the largest rows corresponding to the largest singular values.

Im interested in a nonlinear way to keep all the information too though lol :)

 

[AlephZero]

The only way I have been doing it is with SVD. Once I preform the SVD I rotate the matrix using the V ( from U λ V$^{\dagger}$) and then I project out the largest rows corresponding to the largest singular values.

You might like to think about that idea using the eigenvalues and vectors of the matrix rather than the singular values, to see what it means "physically" for your application.

For Hermitian (or real symmetric) matrices, you can interpret it in terms of partitioning the "energy" of the system and then throwing way the "least important" components.

Of course the SVs and EVs are closely related, and for arbitrary non-hermitian matrices the SVs might be easier to work with.