Is it possible to compare two eigen spaces of different size?
How can i check if there is a substantial change in eigen space of two data matrices, which probably results in different size of eigen space.
Thanks
What you mean by eigenspaces? An operator can have eigenvectors, but what are eigenspaces? If the eigenvalues of the operator are not degenerate, a vector equal to the superposition of two eigenvectors corresponding to two different eigenvalues, is not an eigenvector of the operator.
If A is a principal submatrix (or similar to a principal submatrix) of a matrix B, then every eigenvalues of A is also eigenvalues of B and the eigen space of A belong to an eigenvalue λ is a subspace (up to isomorphic) of the eigen space of B belong to the eigenvalue λ.
@W. Wanicharpichat: This is not true. It only holds if B is (block) triangular and A is one of its diagonal blocks.
An Eigenspaces is the collection of several eigenvectors associated to one or more eigenvalues. A closely related concept is that of invariant subspaces. To compare different eigenspaces, you can use the concept of principal angles of subspaces. See, e.g., https://en.wikipedia.org/wiki/Angles_between_flats#Angles_between_subspaces .
These angles can tell if the subspaces have a nonempty union, i.e., if there are linear dependencies between the basis vectors of both subspaces.
@P. Kürschner, thank you sir. The matrix B much be a block triangular matrix and A is one of its diagonal blocks.
Thanks.. @Patrick: Is angle between different size eigen space is enough? It is not suggesting any change in my data though the change in position is substantial?
Not only the position matters, not only angle.
Is there any combination of position and angle tried by someone to find the how is variation between eigen spaces?
PS: I am taking incremental PCA as soon as new data is added, and i am trying to find how different is it with new increment..
Any Ideas?
Sofia D. Wechsler : Eigenspace is a combination of eigen vectors of data.
I'm affraid i don't fully unterstand where you're going with this.
But there should be possibilities to apply the principal subspace angles also to spaces of different dimensions.
I suggest to look into textbooks covering subspace angle, i.e., more sophisticated sources than wikipedia.
Spatial Efficiency metric (SPAEF) is proven to be robust when comparing two raster maps. Python and Matlab codes are available at: http://space.geus.dk/tools_products/index.html
Two papers on the metric:
1) https://www.hydrol-earth-syst-sci.net/22/1299/2018/hess-22-1299-2018-assets.html
2) https://www.geosci-model-dev-discuss.net/gmd-2017-238/
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